Calculus (3rd Edition)

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

Chapter 3 - Differentiation - 3.5 Higher Derivatives - Exercises - Page 136: 32


$$ f^{(n)}(x)=(-1)^n (n)! \ (x+2)^{-(n+1)}.$$

Work Step by Step

Since $ f(x)=(x+2)^{-1}$, then we have $$ f'(x)=(-1)(x+2)^{-2},\quad f''(x)=(-1)(-2)(x+2)^{-3}, \quad f'''(x)=(-1)(-2)(-3)(x+2)^{-4}, \cdots,$$ Hence, we get the general formula for $ f^{(n)}(x)$ as follows $$ f^{(n)}(x)=(-1)^n (n)! \ (x+2)^{-(n+1)}.$$
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