Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 3 - Derivatives

Problema 3.10.54

Capitolo 3, Problema 3.10.54

47–56. Derivatives of inverse functions at a point Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point.
f(x)=4e^10x; (4,0)

Guida verificata passo dopo passo

Step 1: Understand the problem. We need to find the derivative of the inverse function of f(x) = 4e^{10x} at the point (4, 0). This means we need to find (f^{-1})'(0).

Step 2: Use the formula for the derivative of an inverse function. If y = f(x) and f is invertible, then the derivative of the inverse function at a point is given by (f^{-1})'(y) = 1 / f'(x), where f(x) = y.

Step 3: Identify the point of interest. We are given the point (4, 0), which means f(x) = 4 when x = 0. Therefore, we need to find f'(0).

Step 4: Differentiate the function f(x) = 4e^{10x}. The derivative f'(x) is found using the chain rule: f'(x) = 4 * 10 * e^{10x} = 40e^{10x}.

Step 5: Evaluate the derivative at x = 0. Substitute x = 0 into f'(x) to find f'(0) = 40e^{0} = 40. Now, use the inverse derivative formula: (f^{-1})'(0) = 1 / f'(0) = 1 / 40.

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Inverse Functions

Inverse functions are functions that 'reverse' the effect of the original function. If f(x) takes an input x and produces an output y, then the inverse function f⁻¹(y) takes y back to x. Understanding how to find and work with inverse functions is crucial for evaluating derivatives of inverses.

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Inverse Cosine

Derivative of Inverse Functions

The derivative of an inverse function can be calculated using the formula (f⁻¹)'(y) = 1 / f'(x), where y = f(x). This relationship shows how the rate of change of the inverse function at a point is related to the rate of change of the original function at the corresponding point. This concept is essential for solving problems involving derivatives of inverse functions.

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Exponential functions, such as f(x) = 4e^(10x), have specific properties and derivatives. The derivative of an exponential function is proportional to the function itself, specifically f'(x) = k * e^(kx) for some constant k. Understanding how to differentiate exponential functions is necessary for applying the derivative of the inverse function formula.

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Derivatives of General Exponential Functions

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