The derivative and the power rule
Differentiation finds the rate at which a function changes. The derivative dy⁄dx gives the gradient of the curve y = f(x) at any point. For powers of x the rule is simple.
Key idea
Power rule: if y = xn then dy⁄dx = nxn−1. Constants differentiate to 0, and a constant multiple is kept: d⁄dx (kxn) = knxn−1.
Standard results and rules
You should know: d⁄dx (ex) = ex, d⁄dx (ln x) = 1⁄x, d⁄dx (sin x) = cos x and d⁄dx (cos x) = −sin x. For composite and combined functions use:
- Chain rule: if y = f(u) and u = g(x), then dy⁄dx = (dy⁄du)(du⁄dx).
- Product rule: (uv)′ = u′v + uv′.
- Quotient rule: (u⁄v)′ = (u′v − uv′)⁄v2.
Worked example
Differentiate y = (2x + 1)4. Let u = 2x + 1, so y = u4. Then dy⁄dx = 4u3 × 2 = 8(2x + 1)3.
Remember
- Reduce the power by 1 and multiply by the old power.
- Do not forget the inner derivative in the chain rule.
- The second derivative d2y⁄dx2 is found by differentiating again.