Derivatives of Exponential and Logarithmic Functions

Derivative of $e^x$. By the first-principles definition, and using the standard limit

$\lim_{h\to0}(e^h-1)/h=1$ (established in the limits chapter),

So $e^x$ is its own derivative -- the defining property of the natural exponential, and the

reason $e$ is singled out as the natural base among all possible exponential bases.

Derivative of $\ln x$. Let $y=\ln x$ ($x>0$), so $e^y=x$ by definition (Section 6).

Differentiating both sides implicitly with respect to $x$ (Section 5), and using

$d(e^y)/dx=e^y\,dy/dx$ by the chain rule,

since $e^y=x$. Hence

General exponential base $a^x$. Writing $a^x=e^{x\ln a}$ (the defining relation between a

general base and the natural exponential; take $\ln$ of both sides to see $\ln(a^x)=x\ln a$,

which is exactly $\ln$ of $e^{x\ln a}$), and applying the chain rule with outer function $e^{(\cdot)}$

and inner function $x\ln a$ (whose derivative with respect to $x$ is the constant $\ln a$):

General base logarithm $\log_a x$. Using $\log_a x=\ln x/\ln a$ (Section 6) and that

$1/\ln a$ is a constant,

Combined with the chain rule -- the two most-used composite forms. For a differentiable

function $f(x)$,

These two composite forms -- exponential-of-a-function and log-of-a-function -- appear far more

often in practice than the bare $e^x$ or $\ln x$, and both are direct chain-rule applications of

the two base results above. …