Exponential and Logarithmic Functions

The exponential function \(e^x\) is its own derivative, and the natural logarithm \(\log x\) (base \(e\)) has derivative \(\frac1x\) for \(x>0\). With the chain rule:

\[\frac{d}{dx}e^{g(x)}=e^{g(x)}\,g'(x),\qquad \frac{d}{dx}\log g(x)=\frac{g'(x)}{g(x)},\qquad \frac{d}{dx}a^x=a^x\log a\]

Useful facts: \(\log(ab)=\log a+\log b\), \(\log\frac ab=\log a-\log b\), \(\log a^n=n\log a\). Simplifying the logarithm before differentiating often saves work: \(\frac{d}{dx}\log\frac{x^2}{x+1}=\frac2x-\frac1{x+1}\).