The power rule
The first derivative dy/dx measures the rate at which y changes with respect to x. The basic rule for powers of x is d/dx(xn) = nxn−1. A constant differentiates to 0, and a constant multiple stays outside: d/dx(kxn) = knxn−1. Sums are differentiated term by term.
Key formula
d/dx(xn) = nxn−1 · chain rule: dy/dx = (dy/du)(du/dx) · product: (uv)′ = u′v + uv′
Product, quotient and chain rules
For a product y = uv, use (uv)′ = u′v + uv′. For a quotient y = u/v, use (u/v)′ = (u′v − uv′)/v2. For a composite function such as y = (ax + b)n, use the chain rule: differentiate the outside, then multiply by the derivative of the inside, giving dy/dx = n(ax + b)n−1 × a.
Worked example
Differentiate y = 3x4 − 2x2 + 5. Term by term, dy/dx = 12x3 − 4x. At x = 1, dy/dx = 12 − 4 = 8. For y = (2x + 1)5, the chain rule gives dy/dx = 5(2x + 1)4 × 2 = 10(2x + 1)4.
Negative and fractional powers
Rewrite roots and fractions as powers first. For example 1/x = x−1 so its derivative is −x−2, and √x = x1/2 so its derivative is ½x−1/2 = 1/(2√x).
Remember
- d/dx(xn) = nxn−1 for all n.
- Rewrite roots/fractions as powers before differentiating.
- Use the chain rule for brackets raised to a power.