Calculus, 10th Edition (Anton)

Published by Wiley
ISBN 10: 0-47064-772-8
ISBN 13: 978-0-47064-772-1

Chapter 2 - The Derivative - 2.6 The Chain Rule - Exercises Set 2.6 - Page 158: 12


$f′(x)=\frac{(3x^{2}-2)}{2\sqrt {x^{3}-2x+5}}$

Work Step by Step

Given that $f(x)=\sqrt {x^{3}-2x+5}$ First write it as $f(x)=({x^{3}-2x+5})^{\frac{1}{2}}$ so that it is easier to deal with. Next we write this as $f(x)=(g(x))^\frac{1}{2}$, where $f(x)$ is the outside function and $g(x)={x^{3}-2x+5}$ is the inside function. To find the answer we need to differentiate the outside function $f(x)$ and times it by the derivative of the inside function $g(x)$. $f(x)=({x^{3}-2x+5})^{\frac{1}{2}}$ $f(x)=(g(x))^\frac{1}{2}$ $f′(x)=\frac{1}{2}(g(x))^{\frac{1}{2}-1}×(g′(x))$ $f′(x)=\frac{1}{2}(g(x))^{\frac{-1}{2}}×(3x^{2}-2)$ $f′(x)=\frac{1}{2}({x^{3}-2x+5})^{\frac{-1}{2}}×(3x^{2}-2)$ $f′(x)=\frac{1}{2}(3x^{2}-2)({x^{3}-2x+5})^{\frac{-1}{2}}$ $f′(x)=\frac{(3x^{2}-2)}{2\sqrt {x^{3}-2x+5}}$
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