Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 2 - Differentiation - 2.1 Exercises - Page 103: 19



Work Step by Step

To take the derivative of a function using the limit process, plug into the equation $f'(x)=\lim\limits_{h \to 0}\frac{f(x+h)-f(x)}{h}$ and simplify: $f'(x)=\lim\limits_{h \to 0}\frac{[(x+h)^3-12(x+h)]-(x^3-12x)}{h}$ $f'(x)=\lim\limits_{h \to 0}\frac{x^3+2x^2h+xh^2+hx^2+2xh^2+h^3-12x-12h-x^3+12x}{h}$ $f'(x)=\lim\limits_{h \to 0}\frac{2x^2h+xh^2+hx^2+2xh^2+h^3-12h}{h}$ $f'(x)=\lim\limits_{h \to 0}2x^2+xh+x^2+2xh+h^2-12$ When you can't simplify any further, plug $0$ in for $h$ and simplify: $f'(x)=2x^2+x(0)+x^2+2x(0)+(0)^2-12=2x^2+x^2-12=3x^2-12$
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