The derivative of \(\mathrm{e}^x\) is \(\mathrm{e}^x.\) This is the only function that is its own derivative, and this fact is reason we describe exponential functions as “growing at a rate proportional to their current value.” The derivative of a general exponential function with base \(b\) can be determined from this fact and the chain rule, since \(b^x = \mathrm{e}^{\ln(b)x}.\) Altogether, \[ \frac{\mathrm{d}}{\mathrm{d}x}\biggl( \mathrm{e}^x \biggr) = \mathrm{e}^x \qquad\text{ and }\qquad \frac{\mathrm{d}}{\mathrm{d}x}\biggl( b^x \biggr) = \ln(b)b^x \]
The derivative of \(\ln(x)\) is \(\frac{1}{x}.\) This makes some sense since a logarithm represents iterated division. The derivative of a general logarithmic function with base \(b\) can be determined from this fact and the change-of-base formula for logarithms, \(\log_b(x) = \frac{\ln(x)}{\ln(b)}.\) Altogether \[ \frac{\mathrm{d}}{\mathrm{d}x}\biggl( \ln(x) \biggr) = \frac{1}{x} \qquad\text{ and }\qquad \frac{\mathrm{d}}{\mathrm{d}x}\biggl( \log_b(x) \biggr) = \frac{1}{\ln(b)x} \]