Calculus with Applications (10th Edition)

Published by Pearson
ISBN 10: 0321749006
ISBN 13: 978-0-32174-900-0

Chapter 4 - Calculating the Derivative - 4.4 Derivatives of Exponential Functions - 4.4 Exercises - Page 232: 36


The graphs seem to coincide, illustrating that $\displaystyle \frac{d}{dx}[e^{x}]=e^{x}$

Work Step by Step

On the same coordinate system, graph $f(x)=e^{x}$ and $y=\displaystyle \frac{f(x+h)-f(x)}{h}=\frac{e^{x+00001}-e^{x}}{0.0001}$ The two graphs seem to coincide. $h=0.0001$ is close to 0, so the graphs illustrate that for $f(x)=e^{x},$ $\displaystyle \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}=f(x)$ that is, $\displaystyle \frac{d}{dx}[e^{x}]=e^{x}$
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