EMath 1101 Exponential and Logarithmic Functions

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Questions and Answers

What is the derivative of arcsin(u) with respect to x?

-u' / √(1 - u²)
u' / (1 - u²)
u' / √(1 - u²) (correct)
u' / √(1 + u²)

The function y = arctan(x) has a horizontal asymptote at y = π/2.

False (B)

What is the value of tan(y) when y = arcsec(√5 / 2)?

1/2

The derivative of arccos(u) is _____.

-u' / √(1 - u²)

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Which of the following is the correct simplification of the derivative y' = arcsin(x) + x√(1 - x²)?

2√(1 - x²) (C)

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Match the inverse trigonometric functions with their derivatives:

arccos(u) = -u' / √(1 - u²) arctan(u) = u' / (1 + u²) arcsec(u) = u' / |u|√(u² - 1) arccsc(u) = -u' / |u|√(u² - 1)

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What are the critical numbers for the function y = (arctan x)²?

x = 0

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The graph of y = (arccos x) has a restricted domain of [-1, 1].

True (A)

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What is the correct range for the function y = arcsin x?

-π / 2 ≤ y ≤ π / 2 (B)

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The value of arccos(0) is π/2.

True (A)

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What is the derivative of y = e^x?

e^x

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The equation arctan(2x - 3) = π/4 can be solved by finding that 2x - 3 equals ________.

1

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Match the following inverse trigonometric identities with their corresponding properties:

sin(arcsin x) = x tan(arctan x) = x sec(arcsec x) = x arccos(cos y) = y

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Given y = arcsin(0.3), what is the approximate value of y?

0.305 (C)

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If y = arcsec(√5/2), then tan y equals 1.

False (B)

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If the domain of y = arccos x is -1 ≤ x ≤ 1, then the range of y is ________.

0 ≤ y ≤ π

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Which of the following is the derivative of the function arctan(u)?

u’ / (1 + u^2) (D)

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The derivative of arcsec(u) is given by u’ / |u|√(u^2 - 1).

True (A)

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What is the integral of cosh(u)?

sinh(u) + C

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The derivative of the function sinh(u) is __________.

cosh(u)u'

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Which hyperbolic identity is correct?

cosh^2(x) - sinh^2(x) = 1 (C)

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The derivative of sec(u) includes a negative sign.

False (B)

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What is the result of differentiating the function u^n?

n*u^(n-1)*u'

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Match the following inverse trig functions with their derivatives:

arcsin(u) = u' / √(1 - u^2) arccos(u) = -u' / √(1 - u^2) arctan(u) = u' / (1 + u^2) arccot(u) = -u' / (1 + u^2)

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What is the derivative of the function y = x^x?

xx (1 + ln x) (B)

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The function y = arccos x has a range of [0, π].

True (A)

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What is the value of arctan(√3)?

π/3

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For the function y = arcsin x, the relationship sin(y) = _____ holds true.

x

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Match the following inverse trigonometric functions with their definitions:

y = arcsin x = sin y = x y = arccos x = cos y = x y = arctan x = tan y = x y = arcsec x = sec y = x

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Which property holds true for y = arctan x?

tan(arctan x) = x (B)

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y = arccsc x is defined for all real numbers.

False (B)

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The value of arcsin(-½) is approximately _____ radians.

-π/6

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What is the derivative of the function y = arcsin(2x)?

$\frac{2}{\sqrt{1 - 4x^2}}$ (B)

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The derivative of arcsec u is negative.

False (B)

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The range of the function y = arcsin x is ________.

[-π/2, π/2]

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In analyzing the graph of y = (arctan x)², which of the following points indicates a relative minimum?

x = 0 (B)

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The horizontal asymptote of y = (arctan x)² is π² / 4.

True (A)

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For y = arcsec(√5/2), what is tan y?

1

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What is the derivative of the function y = cosh(u) with respect to x?

sinh(u)u' (C)

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The identity tanh²(x) + sech²(x) = 1 is true.

False (B)

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State the derivative of sin(u).

cos(u)u'

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The derivative of csch(u) is __________.

-(csch(u) coth(u))u'

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Match the following hyperbolic functions with their derivatives:

sinh(u) = cosh(u)u' cosh(u) = sinh(u)u' tanh(u) = sech²(u)u' coth(u) = -(csch²(u))u'

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Which of the following formulas represents the chain rule for the differentiation of e^(u)?

e^u * u' (D)

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The derivative of a constant function is always 1.

False (B)

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What is the result of differentiating |u|?

u'/|u|, u ≠ 0

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What is the first derivative of the function f(x) = (x - 1) cosh x - sinh x?

(x - 1) sinh x + cosh^2 x (C)

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The equation for the shape of a hanging cable is represented by y = a sinh(x/a).

False (B)

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What are the critical numbers for the function f(x) = (x - 1) cosh x - sinh x?

0 and 1

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The second derivative test helps identify the nature of critical points by providing information about the ______.

concavity

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Which of the following statements is true about the exponential function to the base a?

a^x * a^y = a^{x + y} (B)

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The function y = ½^(t / 5715) represents a growth model for carbon-14.

False (B)

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What is the approximate amount of carbon-14 remaining after 10,000 years if the half-life is 5715 years?

0.30 grams

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The logarithmic function to base a is defined as log_a(x) = _____.

1 / (ln(a) * x)

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Match the following properties of exponents with their descriptions:

a^0 = Always equals 1 a^x * a^y = Adds the exponents a^x / a^y = Subtracts the exponents (a^x)^y = Multiplies the exponents

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If a sample contains 1 gram of carbon-14, how much will remain after 2 half-lives?

0.25 grams (D)

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The statement log_a(xy) = log_a(x) + log_a(y) represents a property of logarithms.

True (A)

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What is the value of y when t equals 5715 in the carbon-14 decay model?

0.5 grams

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Which of the following properties of logarithms states that log_a(x/y) equals log_ax minus log_ay?

Quotient Rule (C)

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The derivative of the function a^x is equal to (ln a)a^x for any positive real number a.

True (A)

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What value does log₃1/81 yield?

-4

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The logarithmic function is the inverse of the _____ function.

exponential

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What is the derivative of log₁₀cos x?

-sin x/(ln 10)cos x (A)

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Match the following logarithmic identities with their expressions:

loga1 = 0 logaxy = logax + logay loga(x/y) = logax - logay logaxn = n logax

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What is the result of the derivative d/dx[xⁿ]?

nxn-1

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For any positive base a, a^log_ax equals x for x > 0.

True (A)

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What is the result of evaluating the function $y = (1/2)^{(t/5715)}$ when $t = 5715$?

0.5 grams (C)

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The logarithmic function to a positive base a is defined as $log_a x = 1 / (ln a imes x)$.

False (B)

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What is the half-life of carbon-14?

5715 years

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The property that states a^x / a^y = a^ is known as the property of exponents.

x - y

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Match the following exponential properties with their descriptions:

a^0 = Equal to 1 a^x * a^y = Equal to a^(x+y) (a^x)^y = Equal to a^(xy) a^x / a^y = Equal to a^(x-y)

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Given a sample of carbon-14 that starts at 1 gram, how much will remain after 10,000 years?

0.30 grams (C)

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The function a^x is defined for any positive real number a that is not equal to 1.

True (A)

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What is the formula to express exponential growth when using a base of ½?

y = (1/2)^(t/5715)

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What is the derivative of the function $y = 3^{x}$?

(ln 3)3^{x} (A), 3^{x} ln(3) (C)

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The logarithmic identity $log_{a}(xy) = log_{a}(x) + log_{a}(y)$ is true.

True (A)

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What is the value of $x$ in the equation $log_{2} x = -4$?

1/16

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The property $log_{a}(y) = x$ implies that $y = a^{______}$.

x

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Match the logarithmic property with its description:

$log_{a}(1)$ = 0 $log_{a}(a^{x})$ = x $log_{a}(x/y)$ = $log_{a} x - log_{a} y$ $log_{a}(xy)$ = $log_{a} x + log_{a} y

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Which of the following illustrates the base conversion for logarithms?

$log_{a} x = log_{b} x / log_{b} a$ (C)

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The derivative of the natural logarithm function $log_{e}(x)$ is equal to $1/x$.

True (A)

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Differentiate the function $y = 2^{3x}$ with respect to $x$.

(3 ln 2)2^{3x}

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Which of the following correctly matches the inverse trigonometric functions with their domains?

arctan(x): -∞ < x < ∞ (A), arcsin(x): -1 ≤ x ≤ 1 (B)

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The value of arcsin(1) is π/2.

True (A)

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Evaluate arcsin(0.5).

π/6

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The derivative of y = x^x is __________.

x^x(1 + ln x)

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Match each inverse trigonometric function with its corresponding range:

arcsin(x) = -π/2 ≤ y ≤ π/2 arccos(x) = 0 ≤ y ≤ π arctan(x) = -π/2 < y < π/2 arcsec(x) = 0 ≤ y ≤ π, y ≠ π/2

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What is the value of arccos(-1)?

π (C)

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Arctan(0) is equal to 0.

True (A)

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Using right triangles, if y = arcsin(0.6), find cos(y) given that 0 < y < π/2.

√(1 - (0.6)^2) = √(0.64) = 0.8

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What is the formula for the derivative of a product of two functions, u and v?

u'v + uv' (A)

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The identity tanh²(x) + sech²(x) = 1 is correct.

False (B)

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The derivative of the function y = e^u is ________.

e^u u'

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Match the hyperbolic function with its derivative:

sinh(u) = cosh(u)u' cosh(u) = sinh(u)u' tanh(u) = sech²(u)u' sech(u) = -sech(u)tanh(u)u'

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The function y = a cosh(x/a) describes the shape of a hanging cable and is classified as a parabola.

False (B)

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What is the derivative of cosh(u)?

sinh(u)u' (D)

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Csc(u) has a negative derivative.

True (A)

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The second derivative test is used to determine whether a critical point is a or a .

maximum, minimum

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What is the relationship between the derivatives of the arcsin function?

u' / √(1 - u²)

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For the function y = arcsin(2x), what is the derivative dy/dx?

2 / √(1 - 4x²) (D)

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The derivative of arcsec(u) is positive for all u when u > 1.

False (B)

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What is the value of tan(y) when y = arcsec(√5/2)?

1/2

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Which of the following represents the correct expression for the second derivative of y = (arctan x)²?

2(1 - 2x arctan x)/(1 + x²)² (C)

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The critical number of the function y = (arctan x)² is at x = 0.

True (A)

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What is the formula for the derivative of y = arcsec(u)?

u' / |u|√(u² - 1)

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Flashcards

d/dx[e^x]

The derivative of e^x with respect to x is e^x

d/dx[x^x]

The derivative of x^x with respect to x is xe^x-1

arcsin x

The inverse sine function; sin(arcsin x) = x

arccos x

The inverse cosine function; cos(arccos x) = x

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arctan x

The inverse tangent function; tan(arctan x) = x

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arcsin(-½)

-π/6

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arccos 0

π/2

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arctan √3

π/3

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Derivative of arcsin(u)

The derivative of arcsin(u) with respect to x is u' / √(1 - u²).

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Derivative of arccos(u)

The derivative of arccos(u) with respect to x is -u' / √(1 - u²).

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Derivative of arctan(u)

The derivative of arctan(u) with respect to x is u' / (1 + u²).

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Derivative of arcsec(u)

The derivative of arcsec(u) with respect to x is u' / |u|√(u² - 1).

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Differentiating inverse trig functions

The process of finding the derivative of inverse trigonometric functions (arcsin, arccos, arctan).

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Critical number for y = (arctan x)^2

The x-value that makes the first derivative equal to zero in an equation.

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Horizontal asymptote of y = (arctan x)^2

The horizontal line the graph approaches as x increases or decreases without bound.

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Points of inflection for y=(arctan x)^2

The x values where the graph changes concavity.

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Derivative of sinh(u)

The rate of change of the hyperbolic sine function sinh u with respect to x. Calculated as (cosh u) * u'.

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Derivative of cosh(u)

The rate of change of the hyperbolic cosine function cosh u with respect to x. Calculated as (sinh u) * u'.

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Derivative of tanh(u)

The rate of change of the hyperbolic tangent function tanh u with respect to x. Calculated as (sech² u) * u'.

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Derivative of sech(u)

The rate of change of the hyperbolic secant function sech u with respect to x. Calculated as - (sech u * tanh u) * u'.

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∫cosh u du

The integral of the hyperbolic cosine function cosh u with respect to u. The result is sinh u + C, where C is the constant of integration.

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∫sinh u du

The integral of the hyperbolic sine function sinh u with respect to u. The result is cosh u + C, where C is the constant of integration.

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Hyperbolic Function

Functions that are related to the trigonometric functions but use hyperbolas instead of circles.

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Chain Rule in Differentiation

Used to differentiate composite functions; if y = f(u) and u = g(x), then dy/dx = df/du * du/dx*.

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log_a1

The logarithm of 1 to any base a is always equal to 0.

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log_axy

The logarithm of a product is equal to the sum of the logarithms of the individual factors.

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log_axⁿ

The logarithm of a power is equal to the exponent multiplied by the logarithm of the base.

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log_a(x/y)

The logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator.

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Inverse Functions

Two functions are inverses if, and only if, one function undoes the other's operation. This means that applying both functions in succession results in the original input value.

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d/dx[log_ax]

The derivative of log_ax with respect to x is 1 / (ln a)x.

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What's the derivative of e^x?

The derivative of e^x with respect to x is simply e^x.

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Exponential Function to Base a

A function defined as a^x = e^{(ln a) x}, where a is a positive real number (a ≠ 1) and x is any real number.

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Logarithmic Function to Base a

The inverse of the exponential function to base a, defined as log_ax = 1 / ln(a) x, where a is a positive real number (a ≠ 1) and x is any positive real number.

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Logarithmic Differentiation

A technique used to differentiate functions with complex exponents, like x^x, by taking the natural logarithm of both sides and applying properties of logarithms.

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Radioactive Half-Life

The time it takes for half of a radioactive substance to decay.

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Half-Life Model

A mathematical model that describes the exponential decay of a radioactive substance using the half-life.

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Carbon-14 Dating

A method used to determine the age of organic materials by measuring the amount of carbon-14 remaining.

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Properties of Exponents with Base a

The usual laws of exponents apply to exponential functions with base a. For example:

a⁰ = 1
a^x a^y = a^{x + y}
a^x / a^y = a^{x - y}
(a^x)^y = a^xy

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What is sin(arcsin x)?

If -1 ≤ x ≤ 1, then sin(arcsin x) = x. This means the sine function 'undoes' the arcsine function.

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What is tan(arctan x)?

If -∞ < x < ∞, then tan(arctan x) = x. This means the tangent function 'undoes' the arctangent function.

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How to find the amount of carbon-14 remaining after a given time?

Use the half-life model: y = (½)^{(t/ 5715)}, where y is the amount of carbon-14, t is the time in years, and 5715 is the half-life of carbon-14.

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What is sec(arcsec x)?

If |x| ≥ 1, then sec(arcsec x) = x. This means the secant function 'undoes' the arcsecant function.

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Logarithmic Functions Properties

Logarithmic functions to the base a have similar properties to the natural logarithmic function.

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Derivative of ln(cosh x)

The derivative of ln(cosh x) with respect to x is sinh x / cosh x, which simplifies to tanh x.

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Derivative of x sinh x - cosh x

The derivative of x sinh x - cosh x with respect to x is x cosh x. You can apply the product rule and the derivative of sinh x and cosh x.

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Relative Extrema

Relative extrema are the points on a function's graph where the function reaches a local maximum or minimum value. They are found by setting the first derivative equal to zero and solving for x.

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Catenary

A catenary is the curve formed by a hanging cable or chain suspended between two points. Its equation is given by y = a cosh(x/a), where a is the tension in the cable.

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Critical Number

A critical number is a value in the domain of a function that makes the first derivative equal to zero or undefined.

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Horizontal Asymptote

A horizontal line that the graph of a function approaches as x approaches positive or negative infinity.

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Point of Inflection

A point on a graph where the concavity changes from concave up to concave down, or vice-versa.

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Differentiating Inverse Trigonometric Functions

Finding the derivative of an inverse trigonometric function, like arcsin, arccos, arctan, and so on.

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Constant Multiple Rule

The derivative of a constant times a function is the constant times the derivative of the function.

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Sum/Difference Rule

The derivative of a sum or difference of functions is the sum or difference of their derivatives.

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Product Rule

The derivative of the product of two functions is the first function times the derivative of the second plus the second function times the derivative of the first.

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Quotient Rule

The derivative of the quotient of two functions is equal to the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the denominator squared.

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Power Rule

The derivative of x raised to a power n is n times x raised to the power n-1.

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Derivative of ln(u)

The derivative of the natural logarithm of a function u is the derivative of u divided by u itself.

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Derivative of e^u

The derivative of e raised to a function u is e raised to u multiplied by the derivative of u.

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Derivative of Hyperbolic Sine

The derivative of the hyperbolic sine function sinh u is the hyperbolic cosine of u multiplied by the derivative of u.

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Exponential Rule

The derivative of e^x with respect to x is simply e^x.

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What is log_a1 ?

The logarithm of 1 to any base a is always equal to 0. In other words, a⁰ = 1.

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What is log_axⁿ ?

The logarithm of a power is equal to the exponent multiplied by the logarithm of the base. In other words: log_axⁿ = n *log_a*x.

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Differentiating Composite Functions

Finding the derivative of a function that is made up of two or more functions nested within each other. You need to apply the Chain Rule.

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Chain Rule

Used to differentiate composite functions. If y = f(u) and u = g(x), then dy/dx = df/du * du/dx*.

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Study Notes

EMath 1101 Study Notes

Exponential Function to Base a: If a is a positive real number (a ≠ 1) and x is any real number, the exponential function to base a is defined as a^x = e^{(ln a)x}. If a = 1, then y = 1^x = 1 (a constant function). These functions follow the usual laws of exponents.
Familiar Properties (Exponential Functions to Base a):
- a⁰ = 1
- a^xa^y = a^x+y
- a^x/a^y = a^x-y
- (a^x)^y = a^xy
Radioactive Half-Life Model (Example): Carbon-14 has a half-life of approximately 5715 years. An equation can model the amount of carbon-14 remaining over time (y = (1/2)^t/5715).
Logarithmic Function to Base a: If a is a positive real number (a ≠ 1) and x is any positive real number, the logarithmic function to base a is defined as log_ax = (1/ln a) * ln x.
Properties of Logarithmic Functions to Base a:
- log_a 1 = 0
- log_a(xy) = log_ax + log_ay
- log_a(xⁿ) = n log_ax
- log_a (x/y) = log_a x - log_a y
Exponential and Logarithmic Functions (Inverse Functions): f(x) = a^x and g(x) = log_ax are inverse functions of each other.
Properties of Inverse Functions:
- y = a^x if and only if x = log_ay
- a^log_ax = x (for x > 0)
- log_a(a^x) = x (for all x)
Common Logarithms: The logarithmic function to base 10 is called the common logarithm, denoted as log₁₀x or simply log x.
Derivatives for Bases Other Than e: Rules for differentiating exponential and logarithmic functions with bases other than e are provided involving natural logarithms (ln). These rules are extensions for chain rule.
The Power Rule for Real Exponents: For differentiable function u of x, and real number n
- D(xⁿ)/dx = nx^n-1
- D(uⁿ)/dx = nu^n-1 (du/dx)
Derivatives of Inverse Trigonometric Functions: Formulas are given for differentiating various inverse trigonometric functions like arcsin, arccos, arctan, etc., if u(x) is a differentiable function.
Solving Equations (Bases other than e): Examples demonstrate how to solve equations involving exponential and logarithmic functions using functions with base a.
Evaluating Inverse Trigonometric Functions: Examples show evaluation of inverse trigonometric functions using correct intervals for output values.
Properties of Inverse Trigonometric Functions: Relationships between sine, cosine, tangent and their inverses are detailed.
Differentiation of Inverse Trigonometric Functions (Examples): Specific examples showcase the application of derivative formulas to functions with inverse trigonometric functions.
Analyzing Inverse Trigonometric Graphs: Analysis of graphs for y = (arctan x)² is performed including finding asymptotes, inflection points, and critical points by differentiation.
Hyperbolic Functions: Identities for hyperbolic functions are presented. These include hyperbolic sine (sinh), cosine (cosh), tangent (tanh), and their inverses.
Derivatives and Integrals of Hyperbolic Functions: Formulas for differentiating and integrating hyperbolic functions are detailed.

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EMath 1101 Exponential and Logarithmic Functions

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Questions and Answers

What is the derivative of arcsin(u) with respect to x?

The function y = arctan(x) has a horizontal asymptote at y = π/2.

What is the value of tan(y) when y = arcsec(√5 / 2)?

The derivative of arccos(u) is _____.

Which of the following is the correct simplification of the derivative y' = arcsin(x) + x√(1 - x²)?

Match the inverse trigonometric functions with their derivatives:

What are the critical numbers for the function y = (arctan x)²?

The graph of y = (arccos x) has a restricted domain of [-1, 1].

What is the correct range for the function y = arcsin x?

The value of arccos(0) is π/2.

What is the derivative of y = ex?

The equation arctan(2x - 3) = π/4 can be solved by finding that 2x - 3 equals ________.

Match the following inverse trigonometric identities with their corresponding properties:

Given y = arcsin(0.3), what is the approximate value of y?

If y = arcsec(√5/2), then tan y equals 1.

If the domain of y = arccos x is -1 ≤ x ≤ 1, then the range of y is ________.

Which of the following is the derivative of the function arctan(u)?

The derivative of arcsec(u) is given by u’ / |u|√(u^2 - 1).

What is the integral of cosh(u)?

The derivative of the function sinh(u) is __________.

Which hyperbolic identity is correct?

The derivative of sec(u) includes a negative sign.

What is the result of differentiating the function u^n?

Match the following inverse trig functions with their derivatives:

What is the derivative of the function y = xx?

The function y = arccos x has a range of [0, π].

What is the value of arctan(√3)?

For the function y = arcsin x, the relationship sin(y) = _____ holds true.

Match the following inverse trigonometric functions with their definitions:

Which property holds true for y = arctan x?

y = arccsc x is defined for all real numbers.

The value of arcsin(-½) is approximately _____ radians.

What is the derivative of the function y = arcsin(2x)?

The derivative of arcsec u is negative.

The range of the function y = arcsin x is ________.

In analyzing the graph of y = (arctan x)², which of the following points indicates a relative minimum?

The horizontal asymptote of y = (arctan x)² is π² / 4.

For y = arcsec(√5/2), what is tan y?

What is the derivative of the function y = cosh(u) with respect to x?

The identity tanh²(x) + sech²(x) = 1 is true.

State the derivative of sin(u).

The derivative of csch(u) is __________.

Match the following hyperbolic functions with their derivatives:

Which of the following formulas represents the chain rule for the differentiation of e^(u)?

The derivative of a constant function is always 1.

What is the result of differentiating |u|?

What is the first derivative of the function f(x) = (x - 1) cosh x - sinh x?

The equation for the shape of a hanging cable is represented by y = a sinh(x/a).

What are the critical numbers for the function f(x) = (x - 1) cosh x - sinh x?

The second derivative test helps identify the nature of critical points by providing information about the ______.

Which of the following statements is true about the exponential function to the base a?

The function y = ½^(t / 5715) represents a growth model for carbon-14.

What is the approximate amount of carbon-14 remaining after 10,000 years if the half-life is 5715 years?

The logarithmic function to base a is defined as log_a(x) = _____.

Match the following properties of exponents with their descriptions:

If a sample contains 1 gram of carbon-14, how much will remain after 2 half-lives?

The statement log_a(xy) = log_a(x) + log_a(y) represents a property of logarithms.

What is the value of y when t equals 5715 in the carbon-14 decay model?

Which of the following properties of logarithms states that loga(x/y) equals logax minus logay?

The derivative of the function ax is equal to (ln a)ax for any positive real number a.

What value does log31/81 yield?

The logarithmic function is the inverse of the _____ function.

What is the derivative of log10cos x?

Match the following logarithmic identities with their expressions:

What is the result of the derivative d/dx[xn]?

For any positive base a, alogax equals x for x > 0.

What is the result of evaluating the function $y = (1/2)^{(t/5715)}$ when $t = 5715$?

The logarithmic function to a positive base a is defined as $log_a x = 1 / (ln a imes x)$.

What is the half-life of carbon-14?

The property that states ax / ay = a______ is known as the ______ property of exponents.

Match the following exponential properties with their descriptions:

Given a sample of carbon-14 that starts at 1 gram, how much will remain after 10,000 years?

The function ax is defined for any positive real number a that is not equal to 1.

What is the formula to express exponential growth when using a base of ½?

What is the derivative of the function $y = 3^{x}$?

The logarithmic identity $log_{a}(xy) = log_{a}(x) + log_{a}(y)$ is true.

What is the derivative of y = e^x?

What is the derivative of the function y = x^x?

Which of the following properties of logarithms states that log_a(x/y) equals log_ax minus log_ay?

The derivative of the function a^x is equal to (ln a)a^x for any positive real number a.

What value does log₃1/81 yield?

What is the derivative of log₁₀cos x?

What is the result of the derivative d/dx[xⁿ]?

For any positive base a, a^log_ax equals x for x > 0.

The property that states a^x / a^y = a^ is known as the property of exponents.

The function a^x is defined for any positive real number a that is not equal to 1.

The second derivative test is used to determine whether a critical point is a or a .

d/dx[e^x]

d/dx[x^x]

log_a1

log_axy

log_axⁿ

log_a(x/y)

d/dx[log_ax]

What's the derivative of e^x?