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)=tan x; (1,π/4)

Verified step by step guidance

1. Identify the function given: \( f(x) = \tan x \). We need to find the derivative of its inverse at the point \((1, \pi/4)\).
2. Recall that if \( y = f^{-1}(x) \), then \( f(y) = x \). The derivative of the inverse function at a point \( x = a \) is given by \( (f^{-1})'(a) = \frac{1}{f'(f^{-1}(a))} \).
3. Since \( f(x) = \tan x \), the derivative \( f'(x) = \sec^2 x \).
4. We know \( f(\pi/4) = \tan(\pi/4) = 1 \), so \( f^{-1}(1) = \pi/4 \).
5. Substitute \( f^{-1}(1) = \pi/4 \) into the formula for the derivative of the inverse: \( (f^{-1})'(1) = \frac{1}{f'(\pi/4)} = \frac{1}{\sec^2(\pi/4)} \).

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 that the slope of the tangent line to the inverse function at a point is the reciprocal of the slope of the tangent line to the original function at the corresponding point.

Trigonometric Functions and Their Derivatives

Trigonometric functions, such as f(x) = tan(x), have specific derivatives that are essential for solving problems involving these functions. For example, the derivative of tan(x) is sec²(x). Knowing these derivatives allows for the application of the inverse derivative formula effectively in the context of trigonometric functions.