Explanation Given: g is the inverse of f . Formula used: Chain rule: If y = f ( u ) and u = g ( x ) , then, d y d x = d y d u ⋅ d u d x Calculation: It is given that g is the inverse of f . So, f ( g ( x ) ) = x . Take differentiate on both sides. d d x f ( g ( x ) ) = d d x ( x ) d d x f ( g ( x ) ) = 1 Let u = g ( x ) . Now, use the chain rule. Let y = f ( u ) . y ′ = f ′ ( u ) u ′ … (1) The derivative of u = g ( x ) . u ′ = g ′ ( x ) Substitute the derivative of u ′ in equation (1). y ′ = f ′ ( u ) g ′ ( x ) Substitute the value of u in above equation...

4th Edition

ISBN: 9781319279813

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 3, Problem 28CRE

To determine

Use the chain rule to show that if $g$ is the inverse of $f$ , then $g^{'} (x) = \frac{1}{f^{'} (g (x))}$ for all $x$ in the domain $g$ such that $f (g (x))$ .

Also, use this to obtain another method for finding the derivative of $\ln x$ using the derivative of $e^{x}$ .

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Chapter 3, Problem 27CRE

Chapter 3, Problem 29CRE

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Chapter 3 Solutions

CALCULUS:EARLY TRANS.-SAPLINGPLUS

Ch. 3.1 - Prob. 1PQ Ch. 3.1 - Prob. 2PQ Ch. 3.1 - Prob. 3PQ Ch. 3.1 - Prob. 4PQ Ch. 3.1 - Prob. 5PQ Ch. 3.1 - Prob. 6PQ Ch. 3.1 - Prob. 7PQ Ch. 3.1 - Prob. 1E Ch. 3.1 - Prob. 2E Ch. 3.1 - Prob. 3E

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Use the chain rule to show that if g is the inverse of f , then g ′ ( x ) = 1 f ′ ( g ( x ) ) for all x in the domain g such that f ( g ( x ) ) . Also, use this to obtain another method for finding the derivative of ln x using the derivative of e x .

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Use the chain rule to show that if g is the inverse of f, then g′(x)=1f′(g(x))for all x in the domain gsuch that f(g(x)). Also, use this to obtain another method for finding the derivative of lnx using the derivative of ex.

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Use the chain rule to show that if $g$ is the inverse of $f$ , then $g^{'} (x) = \frac{1}{f^{'} (g (x))}$ for all $x$ in the domain $g$ such that $f (g (x))$ .

Also, use this to obtain another method for finding the derivative of $\ln x$ using the derivative of $e^{x}$ .