Differentiation

The speed of change

• How fast is a rocket accelerating at the exact moment its fuel runs out? How steep is a roller coaster at its highest point?
• Differentiation 微分 answers these questions — it finds the instantaneous rate of change at any point on a curve.

The power rule 幂法则 and chain rule 链式法则

• Differentiation finds the gradient of a curve at each point — the derivative 导数 $f'(x)$ or $\dfrac{dy}{dx}$.
• The power rule: $\dfrac{d}{dx}(x^n) = n x^{n-1}$ (any rational $n$).
• Chain rule for a function inside a function: $\dfrac{d}{dx}f(g(x)) = f'(g(x))\cdot g'(x)$.

The derivative $\dfrac{dy}{dx} = 2x$ gives the gradient at any point: at $x = 2$, the gradient is $4$.

Stationary points 驻点

• A stationary point is where $\dfrac{dy}{dx} = 0$ (the tangent is flat).
• Test with the second derivative 二阶导数:
  • $f''(x) > 0$ → minimum,
  • $f''(x) < 0$ → maximum.
• Also: $\dfrac{dy}{dx} > 0$ → increasing; $< 0$ → decreasing.

At a maximum or minimum the gradient dy/dx is zero

Worked example — finding stationary points

• $y = x^3 - 6x^2 + 9x + 2$.
• $\dfrac{dy}{dx} = 3x^2 - 12x + 9 = 3(x-1)(x-3)$.
• Stationary points at $x = 1$ and $x = 3$.
• $\dfrac{d^2y}{dx^2} = 6x - 12$: at $x=1$, $f'' = -6 < 0$ (maximum); at $x=3$, $f'' = 6 > 0$ (minimum).

Applications of differentiation

• Optimisation 最优化: find the maximum profit or minimum cost by setting $\dfrac{dy}{dx} = 0$.
• Rates of change: velocity is the derivative of displacement; acceleration is the derivative of velocity.
• Curve sketching: stationary points and the sign of $\dfrac{dy}{dx}$ reveal the shape.
• The derivative also gives the gradient of normals (perpendicular to the tangent).

You've got it

Write from a minimum value of -16

• y = x cubed minus 12x.
• Find the gradient at x = 1 and the value at the positive stationary point.