If we write g(x) for ƒ -1(x), Equation (1) can be written as 1 g′(ƒ(a)) = 1/ƒ′(a) , or g′(ƒ(a)) . ƒ′(a) = 1. If we then write x for a, we get g′(ƒ(x)). ƒ′(x) = 1. The latter equation may remind you of the Chain Rule, and indeed there is a connection. Assume that ƒ and g are differentiable functions that are inverses of one another, so that (g ∘ ƒ)(x) = x. Differentiate both sides of this equation with respect to x, using the Chain Rule to express (g ∘ ƒ)′(x) as a product of derivatives of g and ƒ. What do you find?

If we write g(x) for ƒ -1(x), Equation (1) can be written as 1 g′(ƒ(a)) = 1/ƒ′(a) , or g′(ƒ(a)) . ƒ′(a) = 1. If we then write x for a, we get g′(ƒ(x)). ƒ′(x) = 1. The latter equation may remind you of the Chain Rule, and indeed there is a connection. Assume that ƒ and g are differentiable functions that are inverses of one another, so that (g ∘ ƒ)(x) = x. Differentiate both sides of this equation with respect to x, using the Chain Rule to express (g ∘ ƒ)′(x) as a product of derivatives of g and ƒ. What do you find?

Name: If we write g(x) for ƒ -1(x), Equation (1) can be written as 1 g′(ƒ(a)
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Description: If we write g(x) for ƒ -1(x), Equation (1) can be written as 1 g′(ƒ(a)) = 1/ƒ′(a) , or g′(ƒ(a)) . ƒ′(a) = 1. If we then write x for a, we get g′(ƒ(x)). ƒ′(x) = 1. The latter equation may remind you of the Chain Rule, and indeed there is a connection.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section: Chapter Questions

Problem 18T

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