IP AMaths Notes (Upper Sec, Year 3-4): 15) Differentiation Techniques

Beyond single-term powers, you need compound rules to differentiate efficiently.

Rules

• Product: \( \dfrac{\mathrm{d}}{\mathrm{d}x}(uv) = u'v + uv' \).
• Quotient: \( \dfrac{\mathrm{d}}{\mathrm{d}x}\biggl(\dfrac{u}{v}\biggr) = \dfrac{u'v - uv'}{v^2} \) for \( v \neq 0 \).
• Chain: \( \dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{\mathrm{d}y}{\mathrm{d}u} \cdot \dfrac{\mathrm{d}u}{\mathrm{d}x} \).
• Implicit: differentiate both sides treating \( y \) as function of \( x \).

Worked example 1 — Product rule

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

1. Let \( u = 3x^2 - 1 \), \( v = 2x + 5 \).
2. Then \( u' = 6x \), \( v' = 2 \).
3. \( y' = u'v + uv' = 6x(2x + 5) + (3x^2 - 1)(2) \).
4. Expand: \( y' = 12x^2 + 30x + 6x^2 - 2 = 18x^2 + 30x - 2 \).

Worked example 2 — Implicit differentiation

Given \( x^2 + xy + y^2 = 9 \), find \( \dfrac{dy}{dx} \).

1. Differentiate term by term:
   • \( \dfrac{\mathrm{d}}{\mathrm{d}x}(x^2) = 2x \).
   • For \( xy \), use product rule: \( x \dfrac{\mathrm{d}y}{\mathrm{d}x} + y \).
   • \( \dfrac{\mathrm{d}}{\mathrm{d}x}(y^2) = 2y \dfrac{\mathrm{d}y}{\mathrm{d}x} \).
2. Combine: \( 2x + x \dfrac{\mathrm{d}y}{\mathrm{d}x} + y + 2y \dfrac{\mathrm{d}y}{\mathrm{d}x} = 0 \).
3. Group derivatives: \( \bigl(x + 2y\bigr) \dfrac{\mathrm{d}y}{\mathrm{d}x} = -2x - y \).
4. Thus \( \dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{-2x - y}{x + 2y} \).

Try this

Differentiate \( y = \dfrac{\sin x}{x^2} \) and state where the derivative is undefined.