The chain rule differentiates a function of a function: dy/dx = dy/du × du/dx. For y = (x² + 1)³, let u = x² + 1: dy/dx = 3(x² + 1)² × 2x = 6x(x² + 1)².
The product rule: if y = uv, then dy/dx = u dv/dx + v du/dx. The quotient rule: if y = u/v, then dy/dx = (v du/dx − u dv/dx)/v². Learn the standard derivatives too: eᵏˣ → keᵏˣ, ln x → 1/x, sin kx → k cos kx and cos kx → −k sin kx.
Worked example
Differentiate y = x² cos 3x.
- u = x², v = cos 3x: du/dx = 2x and dv/dx = −3 sin 3x.
- dy/dx = x²(−3 sin 3x) + cos 3x (2x).
Answer:2x cos 3x − 3x² sin 3x
Key words
- chain rule
- for a function inside another function
- product rule
- for two functions multiplied
- quotient rule
- for one function divided by another
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