Calculus (3rd Edition)

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

Chapter 3 - Differentiation - 3.3 Product and Quotient Rules - Exercises: 1

Answer

f'(x)=10$x^{4}$+3$x^{2}$

Work Step by Step

The product rule states that if f(x)=h(x)g(x), then f'(x)=g'(x)h(x)+g(x)h'(x). We can find the derivative of the function f(x)=$x^{3}$($2x^{2}+1$) by setting g(x)=$x^{3}$ and h(x)=$2x^{2}+1$, and applying the product rule. f'(x)=$\frac{d}{dx}$[$x^{3}$]($2x^{2}+1$)+($x^{3}$)$\frac{d}{dx}$[$2x^{2}+1$] $\frac{d}{dx}$[$x^{3}$]=3$x^{2}$, using the power rule $\frac{d}{dx}$[$2x^{2}+1$]=4x, using the power rule Therefore, f'(x)=(3$x^{2}$)($2x^{2}+1$)+($x^{3}$)(4x) =(6$x^{4}$+3$x^{2}$)+4$x^{4}$ (Simplify) =10$x^{4}$+3$x^{2}$ (Simplify)
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