​​​​​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)

Verified step by step guidance

1. 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).
2. 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.
3. 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).
4. 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}.
5. 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.

Key Concepts

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.

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.

Exponential Functions and Their Derivatives

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.