Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 6 - Inverse Functions - Review - Exercises - Page 505: 55


$$f^{(n)}(x)=(x+n)e^{x} $$

Work Step by Step

Given $$f(x)= xe^{x},\ \ f^{(n)}(x)=(x+n)e^{x} $$ At $n=1$ $$f'(x)= xe^x+e^x=(x+1)e^{x}$$ Let at $n=k$, $$f^{(k)}(x)=(x+k)e^{x} $$ Differentiate with respect to $x$ \begin{align*} f^{(k+1)}(x)&=(x+k)e^{x}+e^{x}\\ &=(x+k+1)e^{x} \end{align*} Thus is true for $k=n+1$ and $$f^{(n)}(x)=(x+n)e^{x} $$
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