The derivative of $f$ at $x=c$ is the limit of the average rate of change:
$$f'(c)=\lim_{h\to 0}\frac{f(c+h)-f(c)}{h},$$
provided this limit exists. Geometrically it is the slope of the tangent at $\big(c,f(c)\big)$.
Differentiability implies continuity
If $f$ is differentiable at $c$, then it is continuous at $c$. The converse is false: $f(x)=|x|$ is continuous at $0$ but has a corner there, so $f'(0)$ does not exist (left slope $-1$, right slope $+1$). Corners and sharp points break differentiability.
Standard derivatives
| $f(x)$ | $f'(x)$ | $f(x)$ | $f'(x)$ |
|---|---|---|---|
| $x^{n}$ | $nx^{n-1}$ | $\sin x$ | $\cos x$ |
| $\cos x$ | $-\sin x$ | $\tan x$ | $\sec^2 x$ |
| $e^{x}$ | $e^{x}$ | $\ln x$ | $\tfrac1x$ |
The chain rule
For a composite $y=f(g(x))$,
$$\frac{dy}{dx}=f'\big(g(x)\big)\cdot g'(x).$$
Differentiate the outer function, keep the inside, then multiply by the derivative of the inside. Combined with the product rule $(uv)'=u'v+uv'$ and quotient rule, this handles almost every function in the chapter.