2.1 · The derivative: difference quotients and tangent lines

From a secant to a tangent

For a position function $\displaystyle s(t)$, average velocity on $\displaystyle [t,t+h]$ is $\displaystyle \frac{s(t+h)-s(t)}{h}$. This is the slope of a secant line through two points on the position graph. Keep $\displaystyle h\ne0$ while simplifying.

The derivative is the limit of secant slopes as the second input approaches the first. If this finite limit exists, it gives instantaneous velocity or tangent slope. Its units are output units per input unit.

$$f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$$

Evaluate the limit without dividing by zero

For $\displaystyle f(x)=x^2$, expand $\displaystyle (a+h)^2-a^2=2ah+h^2$. For $\displaystyle h\ne0$, division by $\displaystyle h$ gives $\displaystyle 2a+h$. Only after simplification take the limit to obtain $\displaystyle f'(a)=2a$.

The alternate definition $\displaystyle \lim_{x\to a}\frac{f(x)-f(a)}{x-a}$ uses the same secant geometry. Recognize either form when a limit is presented as a derivative.

Use a derivative value in context

A tangent line passes through $\displaystyle (a,f(a))$ with slope $\displaystyle f'(a)$. Use point-slope form rather than substituting the slope as a height. For a height measured in meters over seconds, a derivative value of $\displaystyle 4$ means an instantaneous rise of $\displaystyle 4$ meters per second.

Differentiability implies continuity, but continuity alone is insufficient. The graph of $\displaystyle |x|$ is continuous at $\displaystyle 0$, yet the left slopes approach $\displaystyle -1$ and right slopes approach $\displaystyle 1$. Corners, cusps, vertical tangents, and discontinuities require separate analysis.

$$y-f(a)=f'(a)(x-a)$$