Summary

Continuity at $x=a$: $f(a)$ defined, $\lim_{x\to a}f(x)$ exists, and

$\lim_{x\to a}f(x)=f(a)$ -- equivalently $\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=f(a)$.

Differentiability at $x=a$: $f'(a)=\lim_{h\to0}[f(a+h)-f(a)]/h$ exists. Differentiable $\Rightarrow$ continuous, but NOT conversely ($f(x)=|x|$ is continuous but not differentiable at

$x=0$).

Chain rule: for $y=f(u)$, $u=g(x)$, $\dfrac{dy}{dx}=\dfrac{dy}{du}\cdot\dfrac{du}{dx}$.

Inverse trigonometric derivatives:

Implicit differentiation: differentiate both sides w.r.t. $x$, applying the chain rule to

every $y$-term, then solve algebraically for $dy/dx$.

Exponential/logarithmic functions and derivatives: $e^x>0$ for all $x$, $\ln x$ defined for

$x>0$ as the inverse of $e^x$; $\dfrac{d}{dx}e^x=e^x$, $\dfrac{d}{dx}\ln x=\dfrac1x$,

$\dfrac{d}{dx}a^x=a^x\ln a$, $\dfrac{d}{dx}\log_ax=\dfrac{1}{x\ln a}$; composite forms

$\dfrac{d}{dx}e^{f(x)}=e^{f(x)}f'(x)$, $\dfrac{d}{dx}\ln f(x)=\dfrac{f'(x)}{f(x)}$.

Logarithmic differentiation: for $y=[f(x)]^{g(x)}$ (variable base AND exponent) or a lengthy

product/quotient, take $\ln$ of both sides first, differentiate implicitly, then multiply back by

$y$: $\dfrac{dy}{dx}=y\cdot\dfrac{d}{dx}\big[\ln y\big]$.

Parametric derivatives: for $x=f(t),\,y=g(t)$, $\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}$

($dx/dt\neq0$); the second derivative is …