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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)?
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?
Which rule applies correctly to the expression 𝑎^(𝑥+y)?
Which rule applies correctly to the expression 𝑎^(𝑥+y)?
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?
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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?
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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)?
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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)?
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If $y = rac{1}{5} x^{10}$, what is $y'$?
If $y = rac{1}{5} x^{10}$, what is $y'$?
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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 $c imes f(x)$?
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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 $y'$?
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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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Description
Test your understanding of derivatives and their applications in exponential functions. This quiz covers various concepts, including the differentiation of log functions and the nature of exponential function ranges. Perfect for calculus students looking to reinforce their knowledge.
