​​​​​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.