Overview
Differentiation finds the gradient of a curve at any point — the rate of change.
The power rule
If y = xⁿ then dy/dx = n xⁿ⁻¹. A constant differentiates to 0; constants multiply through (d/dx of a xⁿ = a n xⁿ⁻¹).
The other rules
- Chain rule: for y = f(g(x)), dy/dx = f′(g(x)) × g′(x).
- Product rule: (uv)′ = u′v + uv′.
- Quotient rule: (u/v)′ = (u′v − uv′) ÷ v².
Standard results
- d/dx (sin x) = cos x, d/dx (cos x) = −sin x
- d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x
Worked example
y = 3x⁴ − 2x + 5 → dy/dx = 12x³ − 2.
Common mistakes
- Forgetting the chain rule for something like (2x + 1)⁵.
Exam tips
- The gradient of a curve at a point = substitute the x-value into dy/dx.