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Questions and Answers
What is the derivative of the function defined as 𝑓(𝑥) = log(𝑥) − log(1 + 𝑥²)?
What is the derivative of the function defined as 𝑓(𝑥) = log(𝑥) − log(1 + 𝑥²)?
- −x/(1 + 𝑥²)ln10
- −(2𝑥²)/(𝑙𝑛10(1 + 𝑥²))
- −(2𝑥)/(𝑥(1 + 𝑥²)ln10) (correct)
- −(2𝑥)/(𝑙𝑛10(1 + 𝑥²))
Which of the following statements is true regarding exponential functions with base 𝑎?
Which of the following statements is true regarding exponential functions with base 𝑎?
- The function is defined for all real numbers. (correct)
- The domain is limited to positive real numbers.
- The base 𝑎 can be any real number.
- The range includes negative real numbers.
Which rule applies correctly to the expression 𝑎^(𝑥+y)?
Which rule applies correctly to the expression 𝑎^(𝑥+y)?
- 𝑎^(𝑥+y) = 𝑎^x × 𝑎^y (correct)
- 𝑎^x + 𝑎^y = 𝑎^(𝑥+y)
- 𝑎^(𝑥+y) = 𝑎^x ÷ 𝑎^y
- 𝑎^(𝑥+y) = (𝑎^y)^x
For the function 𝑓(𝑥) = 𝑎^𝑥, what is the nature of its range?
For the function 𝑓(𝑥) = 𝑎^𝑥, what is the nature of its range?
When differentiating the function 𝑓(𝑥) = log₂(𝑥⁴ + 9), what is the result?
When differentiating the function 𝑓(𝑥) = log₂(𝑥⁴ + 9), what is the result?
What is the slope of the tangent line to the curve $y = x^4$ at the point (1,1)?
What is the slope of the tangent line to the curve $y = x^4$ at the point (1,1)?
Which of the following correctly represents the equation of the tangent line to the curve $y = x^4$ at the point (1,1)?
Which of the following correctly represents the equation of the tangent line to the curve $y = x^4$ at the point (1,1)?
If $y = rac{1}{5} x^{10}$, what is $y'$?
If $y = rac{1}{5} x^{10}$, what is $y'$?
When using the constant multiple rule for differentiation, if $f(x)$ is differentiable and $c$ is a constant, what is the derivative of $c imes f(x)$?
When using the constant multiple rule for differentiation, if $f(x)$ is differentiable and $c$ is a constant, what is the derivative of $c imes f(x)$?
For the function $y = rac{4}{x^{4}}$, what is the value of $y'$?
For the function $y = rac{4}{x^{4}}$, what is the value of $y'$?
Flashcards
Derivative of log(x^4+9)
Derivative of log(x^4+9)
The derivative of log base 2 of (x^4 + 9) is (4x^3)/(ln2(x^4+9))
Derivative of tan⁻¹(lnx)
Derivative of tan⁻¹(lnx)
Derivative of tan⁻¹(lnx) is 1/([1+(lnx)²]x) via chain rule
Derivative of log(1+x²) using log rules
Derivative of log(1+x²) using log rules
The derivative of log(1+x²) = (1 - 2x²)/(x(1+x²))ln(10)
Exponential function rule: a^(x+y)
Exponential function rule: a^(x+y)
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Exponential function rule: (a^x)^y
Exponential function rule: (a^x)^y
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Tangent Line Equation
Tangent Line Equation
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Derivative of a Constant Multiple
Derivative of a Constant Multiple
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Derivative of x^n (Power Rule)
Derivative of x^n (Power Rule)
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Slope of Tangent Line
Slope of Tangent Line
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Equation of a Tangent Line
Equation of a Tangent Line
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Study Notes
Chapter 1: Differentiation
- The derivative is crucial in various branches of mathematics. Formulas for derivatives of different functions are essential.
- Definition: The derivative of a function f at x = a (f'(a)) is the limit as x approaches a of [f(x) - f(a)] / (x - a), provided the limit exists. Geometrically, this represents the slope of the tangent line at point (a, f(a)).
- Alternatively, the derivative of f at x = a is the limit as h approaches 0 of [f(a + h) - f(a)] / h, if the limit exists.
- The derivative can also be viewed as a function of x, defined by adapting the second limit definition for all values of x where the limit exists.
Rule 1: Derivative of a Constant
- The derivative of a constant function is 0.
Rule 2: Derivative of the Identity Function
- The derivative of f(x) = x is 1.
Rule 3: Power Rule for Positive Integers
- The derivative of xn, where n is a positive integer, is nx(n-1).
Rule 4: Constant Multiple Rule
- The derivative of cf(x), where c is a constant, is c f'(x).
Rule 5: Sum Rule for Differentiation
- The derivative of (f(x) + g(x)) is f'(x) + g'(x).
Other Rules
- Chain Rule: For composite functions f(g(x)), the derivative is f'(g(x)) * g'(x).
- Product Rule: For the product of two functions f(x)g(x), the derivative is f'(x)g(x) + f(x)g'(x).
- Quotient Rule: For the quotient of two functions f(x)/g(x), the derivative is [f'(x)g(x) - f(x)g'(x)] / [g(x)]².
Chapter 3: The Chain Rule
- The chain rule is a powerful differentiation technique for composite functions.
- For functions y=f(u) and u=g(x), the derivative of the composition f(g(x)) is f'(g(x)) * g'(x).
- The chain rule is also applicable when the composite function consists of a longer chain of functions.
Derivatives of Trig Functions
- d/dx (sinx) = cosx
- d/dx (cosx) = -sinx
- d/dx (tanx) = sec²x
- d/dx (cotx) = -csc²x
- d/dx (secx) = secx tanx
- d/dx (cscx) = -cscx cotx
Chapter 5: Differentiation of Inverse Trig Functions
- d/dx (sin⁻¹ x) = 1 / √(1-x²)
- d/dx (cos⁻¹ x) = -1 / √(1-x²)
- d/dx (tan⁻¹ x) = 1 / (1 + x²)
- d/dx (cot⁻¹ x) = -1 / (1 + x²)
- d/dx (sec⁻¹ x) = 1 / |x|√(x² - 1)
- d/dx (csc⁻¹ x) = -1 / |x|√(x² - 1)
Chapter 6: Differentiation of Logarithmic and Exponential Functions
- Derivatives of logarithmic functions:
- d/dx (logₐ x) = 1/(x ln a)
- d/dx (ln x) = 1/x
- Derivatives of exponential functions:
- d/dx (aˣ) = aˣ ln a
- d/dx (eˣ) = eˣ
Chapter 7: Differentiation of Hyperbolic Functions
- Derivatives of hyperbolic functions:
- d/dx (sinh x) = cosh x
- d/dx (cosh x) =sinh x
- d/dx (tanh x) = sech²x
- d/dx (coth x) = -csch²x
- d/dx (sech x) = -sech x tanh x
- d/dx (csch x) = -csch x coth x
Chapter 8: Implicit Differentiation
- Implicit differentiation is a technique for finding the derivative of a function when it's not explicitly defined as y = f(x). Treat y as a function of x, and use the chain rule to differentiate both sides of the equation with respect to x.
Chapter 9: Higher-Order Derivatives
- Higher-order derivatives are derivatives of higher orders (second derivative, third derivative, etc.).
- To find the nth derivative, differentiate the function repeatedly.
- Basic derivative patterns for repeatedly differentiating trigonometric functions and exponential functions exist
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