Differentiation — H2 Maths help

Differentiation measures how a function changes. In H2 Maths (syllabus 9758) you use it for gradients, tangents and normals, stationary points, and connected rates of change.

What you need to know

Power rule: if $y = x^n$ then $\dfrac{dy}{dx} = n x^{n-1}$ .
Chain rule: if $y = f(g(x))$ then $\dfrac{dy}{dx} = f'(g(x))\,g'(x)$ .
Product rule: $\dfrac{d}{dx}(uv) = u\dfrac{dv}{dx} + v\dfrac{du}{dx}$ .
Quotient rule: $\dfrac{d}{dx}\!\left(\dfrac{u}{v}\right) = \dfrac{v\,u' - u\,v'}{v^2}$ .

Worked example 1 — chain rule

Differentiate $y = (3x^2 + 1)^5$ .

Let $u = 3x^2 + 1$ , so $y = u^5$ and $\dfrac{du}{dx} = 6x$ .

\frac{dy}{dx} = 5u^4 \cdot \frac{du}{dx} = 5(3x^2+1)^4 \cdot 6x = 30x(3x^2+1)^4.

Worked example 2 — product rule

Differentiate $y = x^2 e^{x}$ .

With $u = x^2$ and $v = e^x$ : $u' = 2x$ , $v' = e^x$ .

\frac{dy}{dx} = u'v + uv' = 2x e^x + x^2 e^x = x e^x (2 + x).

Common mistakes

Forgetting the inner derivative in the chain rule (the $g'(x)$ factor).
Mixing up the order in the quotient rule numerator — it is $v u' - u v'$ , not $u v' - v u'$ .
Treating $e^{x}$ as if $\dfrac{d}{dx}e^x = x e^{x-1}$ — it is $e^x$ .