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}
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\[\frac{d(e^u)}{dx}=e^u\left(\frac{du}{dx}\right)\] |
Example 1: Find the derivative of \[s=2^{2t}\sin 2t\]
Solution:
\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 \[I=\ln\sin 2e^{6t}\]
Solution:
\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}
