Applications of Differentiation

At point $(a, f(a))$:

• Tangent: $y - f(a) = f'(a)(x - a)$
• Normal: $y - f(a) = -\frac{1}{f'(a)}(x - a)$

Set $f'(x) = 0$ to find stationary points.

Classification

Second derivative test:

• $f''(x) > 0$: minimum
• $f''(x) < 0$: maximum
• $f''(x) = 0$: use sign change of $f'(x)$

$f'(x) > 0$: increasing. $f'(x) < 0$: decreasing.

1. Form an expression for the quantity to maximise/minimise.
2. Differentiate and set $= 0$.
3. Solve and verify (max or min).

$\frac{dV}{dt} = \frac{dV}{dr} \times \frac{dr}{dt}$ (chain rule with time).

$f(x) = x^3 - 3x^2 + 4$. $f'(x) = 3x^2 - 6x = 3x(x-2)$.

Stationary at $x = 0$ and $x = 2$.

$f''(0) = -6 < 0$ → max at $(0, 4)$. $f''(2) = 6 > 0$ → min at $(2, 0)$.