Skip to Main Content
It looks like you're using Internet Explorer 11 or older. This website works best with modern browsers such as the latest versions of Chrome, Firefox, Safari, and Edge. If you continue with this browser, you may see unexpected results.


Exponential Derivatives

We can use the logarithmic derivatives to show the derivative of exponential functions. Consider the function \(y=b^x\),

\begin{align} \ln y &=\ln b^x \\ lny &= x\ln b \\ \frac{1}{y} \frac{dy}{dx} &= \ln b \\ \frac{dy}{dx} &=y\ln b \\ \frac{dy}{dx}&=b^x\ln b \end{align}

\[\frac{d(b^u)}{dx}=b^u\ln b\left(\frac{du}{dx}\right)\]

For the exponential function with base \(e\),

\begin{align} y&=e^x \\ \frac{dy}{dx} &= e^x\ln e \\ &=e^x\end{align}


Example 1: Find the derivative of \[s=2^{2t}\sin 2t\]


\begin{align} s'&=2^{2t}\ln 2(2)(\sin 2t)+2\cos 2t (2^{2t}) \\ &=2^{2t+1}\ln 2(\sin 2t) +2^{2t+1}\cos 2t \end{align}

Example 2: Find the derivative of \[I=\ln\sin 2e^{6t}\]


\begin{align} I'&=\left(\frac{1}{\sin 2e^{6t}} \right)\left(\cos 2e^{6t} \right)\left(12e^{6t} \right) \\ &=\frac{12e^{12t}\cos 2e^{6t}}{\sin 2e^{6t}}  \end{align}

Creative Commons License
Designed by Matthew Cheung. This work is licensed under a Creative Commons Attribution 4.0 International License.
chat loading...