Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 3 - Differentiation - 3.7 The Chain Rule - Exercises: 37

Answer

y'=$\frac{-sec(\frac{1}{x})tan(\frac{1}{x})}{x^{2}}$

Work Step by Step

The chain rule states that if y=g(h(x)), then y'=g'(h(x))h'(x). Since y=sec($\frac{1}{x}$), we can set g(x)=sec(x) (outside function) and h(x)=$\frac{1}{x}$ (inside function), and use the chain rule to find y'. Therefore, y'=g'($\frac{1}{x}$)h'(x) We know that $\frac{d}{dx}$[sec(x)]=sec(x)tan(x), which can be determined by using the quotient rule to compute the derivative of the function g(x)=$\frac{1}{cos(x)}$, which of course is equivalent to sec(x) $\frac{d}{dx}$[$\frac{1}{x}$]=-1$x^{-2}$, using the power rule (rewrite $\frac{1}{x}$ as $x^{-1}$) Now that we know that g'(x)=sec(x)tan(x) and h'(x)=-1$x^{-2}$, we can find y'. y'=sec($\frac{1}{x}$)tan($\frac{1}{x}$)*-1$x^{-2}$ =$\frac{-sec(\frac{1}{x})tan(\frac{1}{x})}{x^{2}}$
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