Find the derivative of log (e^x + (1+e^2x)^1/2)

by RoRi

November 13, 2015

Find the derivative of the following function:

$f(x) = \log \left( e^x + \sqrt{1+e^{2x}} \right).$

We compute using the chain rule and the formulas for derivatives of logarithms and exponentials,

$\begin{align*} f'(x) &= \left( \frac{1}{e^x + \sqrt{1+e^{2x}}} \right) \left( e^x + \frac{e^{2x}}{\sqrt{1+e^{2x}}} \right) \\[9pt] &= \frac{ e^x \sqrt{1+e^{2x}} + e^{2x}}{\left( e^x + \sqrt{1+e^{2x}} \right) \left( \sqrt{1+e^{2x}} \right)} \\[9pt] &= \frac{e^x (e^x + \sqrt{1+e^{2x}} )}{\left( e^x + \sqrt{1+e^{2x}} \right) \left( \sqrt{1+e^{2x}} \right)} \\[9pt] &= \frac{e^x}{\sqrt{1+e^{2x}}} \\[9pt] &= e^x \left( 1+ e^{2x} \right)^{-\frac{1}{2}}. \end{align*}$

Stumbling Robot

Find the derivative of log (e^x + (1+e^2x)^1/2)

Related

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment): Cancel reply