Calculus (3rd Edition)

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

Chapter 3 - Differentiation - Chapter Review Exercises - Page 162: 12

Answer

$$\frac{-2}{(x-1)^{3}}$$

Work Step by Step

Given $$y=\frac{1}{(x-1)^{2}}$$ Now we find the derivative with limits: \begin{align*} \frac{d y}{d x}&=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\\ &=\lim _{h \rightarrow 0} \frac{\frac{1}{((x+h)-1)^{2}}-\frac{1}{(x-1)^{2}}}{h}\\ &=\lim _{h \rightarrow 0} \frac{\frac{(x-1)^{2}-((x+h)-1)^{2}}{(x-1)^{2}((x+h)-1)^{2}}}{h}\\ &=\lim _{h \rightarrow 0} \frac{\left(x^{2}-2 x+1\right)-\left((x+h)^{2}-2(x+h)+1\right)}{h(x-1)^{2}((x+h)-1)^{2}}\\ &=\lim _{h \rightarrow 0} \frac{\left(x^{2}-2 x+1\right)-\left(x^{2}+2 x h+h^{2}-2 x-2 h+1\right)}{h(x-1)^{2}((x+h)-1)^{2}}\\ &=\lim _{h \rightarrow 0} \frac{-2 x-h+2}{(x-1)^{2}((x+h)-1)^{2}}\\ &=\frac{-2(x-1)}{(x-1)^{4}}\\ &=\frac{-2}{(x-1)^{3}} \end{align*}
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