Calculus

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

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

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

For the exponential function with base $e$,

$$\begin{array}{rl}y& =e^x\\ \frac{dy}{dx}& =e^x\ln e\\ & =e^x\end{array}$$

$$\frac{d(e^u)}{dx}=e^u\left(\frac{du}{dx}\right)$$

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

Solution:

$$\begin{array}{rl}{s}^{\prime }& =2^{2t}\ln 2(2)(\sin 2t)+2\cos 2t(2^{2t})\\ & =2^{2t+1}\ln 2(\sin 2t)+2^{2t+1}\cos 2t\end{array}$$

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

Solution:

$$\begin{array}{rl}{I}^{\prime }& =\left(\frac{1}{\sin 2e^{6t}}\right)(\cos 2e^{6t})\left(12e^{6t}\right)\\ & =\frac{12e^{12t}\cos 2e^{6t}}{\sin 2e^{6t}}\end{array}$$

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