Question

Suppose you know that f(x) is a one-to-one function whose derivative is f′(x). Find a formula for the derivative of the inverse function f⁻¹(x) in terms of

Answer 1

To find the derivative of the inverse function f⁻¹(x), use the chain rule and take the reciprocal of the derivative of f(x).

Step-by-step explanation:

To find the formula for the derivative of the inverse function f⁻¹(x), we can use the chain rule. Let's denote the inverse function as g(x) = f⁻¹(x). The derivative of g with respect to x, or g'(x), is equal to 1 divided by the derivative of f with respect to x, or f'(x). In equation form, g'(x) = 1 / f'(x).

For example, if f(x) = sin(x), then f'(x) = cos(x). To find g'(x), we take the reciprocal of f'(x): g'(x) = 1 / cos(x). So the derivative of the inverse sine function is 1 / cos(x), or sec(x).