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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 3 - DerivativesProblema 3.10.51
Capitolo 3, Problema 3.10.51

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)

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

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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 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.
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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.
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Introduction to Trigonometric Functions
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