Differentiation
The rules
- Differentiation finds the gradient of a curve at each point — the derivative $f'(x)$ 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: $\dfrac{d}{dx}f(g(x)) = f'(g(x))\cdot g'(x)$.
Practice
If y = x³, then dy/dx = 3x². What is the gradient at x = 2?
dy/dx = 3x² = 3 × 2² = 3 × 4 = 12.
Stationary points
- A stationary point is where $\dfrac{dy}{dx} = 0$ (the tangent is flat).
- Test with the second derivative:
- $f''(x) > 0$ → minimum,
- $f''(x) < 0$ → maximum.
- Also: $\dfrac{dy}{dx} > 0$ → increasing; $< 0$ → decreasing.
Practice
The curve y = x² − 6x + 5 has a stationary point. At what x value?
dy/dx = 2x − 6 = 0 gives x = 3.
Practice
At a stationary point, if the second derivative f″(x) > 0, the point is a:
f″ > 0 means the curve bends upward — a minimum; f″ < 0 gives a maximum.
You've got it
Key idea
- power rule: $\dfrac{d}{dx}(x^n) = n x^{n-1}$; chain rule for nested functions
- stationary point: $\dfrac{dy}{dx} = 0$
- second-derivative test: $f'' > 0$ minimum, $f'' < 0$ maximum