The derivative is the limit from Unit 1 put to work: the instantaneous rate of change, the slope of the tangent line. This unit defines it, says when it exists, and builds the rules that make finding it fast.
The difference quotient $\frac{f(a+h) - f(a)}{h}$ (or $\frac{f(x) - f(a)}{x - a}$) is an average rate of change over an interval. Its limit as $h \to 0$ is the instantaneous rate of change at $a$, written $f'(a)$, when that limit exists.
The derivative function is $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$, also written $\frac{dy}{dx}$ or $y'$. Its value at a point is the slope of the tangent line there, so the tangent line at $x = a$ is $y = f(a) + f'(a)(x - a)$.
From a table, estimate $f'(a)$ with the slope between nearby points; for an interior point, the symmetric difference quotient $\frac{f(a+h) - f(a-h)}{2h}$ is usually best. From a graph, estimate the slope of the tangent line. Always attach units.
If $f$ is differentiable at $a$, it is continuous at $a$. The reverse is false: a continuous function can fail to be differentiable at a corner (like $|x|$ at $0$), a cusp, or a vertical tangent (like $\sqrt[3]{x}$ at $0$). A discontinuity always means no derivative.
Power rule: $\frac{d}{dx}x^n = nx^{n-1}$ for any real $n$. Constants differentiate to $0$, constant multiples come along for the ride, and sums and differences differentiate term by term, which together handle every polynomial.
$\frac{d}{dx}\sin x = \cos x$, $\frac{d}{dx}\cos x = -\sin x$, $\frac{d}{dx}e^x = e^x$, $\frac{d}{dx}\ln x = \frac{1}{x}$. A limit that looks like a difference quotient can sometimes be read as one of these derivatives.
Product rule: $(fg)' = f'g + fg'$. Quotient rule: $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$. With them: $\frac{d}{dx}\tan x = \sec^2 x$, $\frac{d}{dx}\cot x = -\csc^2 x$, $\frac{d}{dx}\sec x = \sec x\tan x$, $\frac{d}{dx}\csc x = -\csc x\cot x$.