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 - Page 146: 12



Work Step by Step

The chain rule states that if y=g(h(x)), then y'=g'(h(x))h'(x). Since y=$(8x^{4}+5)^{3}$, we can set g(x)=$x^{3}$ (outside function) and h(x)=8$x^{4}$+5 (inside function), and use the chain rule to find y'. Therefore, y'=g'(8$x^{4}$+5)h'(x) We know that $\frac{d}{dx}$[$x^{3}$]=3$x^{2}$, using the power rule $\frac{d}{dx}$[8$x^{4}$+5]=32$x^{3}$, using the power rule Now that we know that g'(x)=3$x^{2}$ and h'(x)=4$x^{3}$, we can find y'. y'=3$(8x^{4}+5)^{2}$(32$x^{3}$) =96$x^{3}$$(8x^{4}+5)^{2}$
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