Differentiation
Differentiation
Differentiation questions test technique and interpretation: the derivative as a gradient, the chain, product and quotient rules, derivatives of exponential, logarithmic and trigonometric functions, and applications to tangents, normals, stationary points and rates of change.
Core rules
- d/dx(x^n) = nx^(n−1); d/dx(e^x) = e^x; d/dx(ln x) = 1/x; d/dx(sin x) = cos x; d/dx(cos x) = −sin x; d/dx(tan x) = sec²x.
- Chain rule: dy/dx = dy/du × du/dx. Product: (uv)′ = u′v + uv′. Quotient: (u/v)′ = (u′v − uv′)/v².
- Implicit: differentiate each term with respect to x, remembering that y-terms pick up a factor dy/dx.
- Parametric: dy/dx = (dy/dt)/(dx/dt).
- Stationary points occur where dy/dx = 0; classify by the sign of d²y/dx² (positive minimum, negative maximum) or by a sign table.
Worked example
Differentiate y = x²e^(3x). This is a product with u = x² and v = e^(3x). Then u′ = 2x and, by the chain rule, v′ = 3e^(3x). So dy/dx = 2x e^(3x) + x²(3e^(3x)) = x e^(3x)(2 + 3x). Factorising is worth a mark and makes the stationary points x = 0 and x = −2/3 obvious.
Traps
- Differentiating e^(3x) as e^(3x) rather than 3e^(3x).
- Reversing the order in the quotient rule, which flips the sign.
- Forgetting dy/dx when differentiating a y-term implicitly.
- Applying the second derivative test where it equals zero and drawing a conclusion anyway.