Q.Differentiate $y=e^{3x^2}$ with respect to $x$.

From the standard limit $\lim_{h\to0}(e^h-1)/h=1$, first principles give $\frac{d}{dx}e^x=e^x$;

implicit differentiation of $e^y=x$ then gives $\frac{d}{dx}\ln x=1/x$. Writing a general base as

$a^x=e^{x\ln a}$ extends these via the chain rule to $\frac{d}{dx}a^x=a^x\ln a$ and

$\frac{d}{dx}\log_ax=1/(x\ln a)$. The two composite forms used constantly in practice are

$\frac{d}{dx}e^{f(x)}=e^{f(x)}f'(x)$ and $\frac{d}{dx}\ln f(x)=f'(x)/f(x)$, both direct

chain-rule applications of the base results.