Differentiation of Standard Functions

The following standard differentiation formulae are used directly, as given rules, throughout this chapter:

A constant function never changes, so its rate of change is $0$. The power rule ($x^n \to n x^{n-1}$) is the direct differentiation counterpart of §4's standard algebraic limit formula. The results for $e^x$ and $\ln x$ trace directly back to §5's exponential and logarithmic standard limits — in particular, $e^x$ is the unique function that is its own derivative, which is exactly why it is central to continuous-growth business models (continuous compounding, continuous depreciation). The general exponential rule $a^x \to a^x \ln a$ reduces to the $e^x$ case when $a=e$ (since $\ln e = 1$). …