Unit content
Introduction to differentiation
A function can change at different rates at different points. The derivative measures its instantaneous rate of change.
For a small change $h$, the quotient
$$\frac{f(a+h)-f(a)}{h}$$
is the average rate of change between $a$ and $a+h$. Letting $h$ approach zero gives the derivative at $a$:
$$f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}.$$
Tangent slope
On the graph $y=f(x)$, the same quotient is the slope of a secant line through two nearby points. As the second point approaches the first, the secant slope approaches the slope of the tangent line.
Thus $f'(a)$ has two connected interpretations: instantaneous rate of change and tangent-line slope.
Example from the definition
For $f(x)=x^2$,
$$\frac{f(a+h)-f(a)}{h} =\frac{(a+h)^2-a^2}{h} =2a+h.$$
Taking $h\to0$ gives
$$f'(a)=2a.$$
So the slope of the parabola changes from point to point.
Differentiability
A function is differentiable at $a$ when this limit exists as a finite real number. Corners, cusps, jumps and vertical tangents can prevent differentiability.
Differentiability is stronger than continuity: if a function is differentiable at a point, then it is continuous there.