Summary

Rate of change. If $y = f(x)$ and both vary with time $t$, $\dfrac{dy}{dt} = f'(x)\cdot\dfrac{dx}{dt}$ (related rates, via the chain rule); a positive rate means increasing, a negative rate means decreasing.

Increasing/decreasing. With $f$ continuous on $[a,b]$, differentiable on $(a,b)$: $f'(x) > 0$ throughout $\Rightarrow$ increasing; $f'(x) < 0$ throughout $\Rightarrow$ decreasing; $f'(x) = 0$ throughout $\Rightarrow$ constant. Intervals are found by solving $f'(x) = 0$ for critical points, then sign-testing $f'$ on each resulting interval.

Tangent and normal to $y = f(x)$ at $(x_0, y_0)$:

If $f'(x_0)=0$: the tangent is $y=y_0$ (horizontal) and the normal is $x=x_0$ (vertical).

Local maxima/minima -- critical points. Occur only where $f'(c)=0$ or $f'(c)$ fails to exist.

First derivative test (geometric): $f'$ changes $+\to-$ at $c$ $\Rightarrow$ local max; $-\to+$ $\Rightarrow$ local min; no sign change $\Rightarrow$ neither.

Second derivative test (given as a provable tool, via concavity): at a critical point with $f'(c)=0$: $f''(c)<0 \Rightarrow$ local max; $f''(c)>0 \Rightarrow$ local min; $f''(c)=0 \Rightarrow$ inconclusive, use the first derivative test instead. …