Rules for derivatives

Sum and difference: (u ± v)′ = u′ ± v′.

Product (Leibnitz) rule: (uv)′ = u′v + uv′. A constant factor comes out: (cu)′ = cu′.

Quotient rule: (u/v)′ = (u′v − uv′)/v², where v ≠ 0.

From first principles the derivative of x is 1. Then 10x can be done two ways: as x + x + … (ten terms) by the sum rule, or as 10 · x by the product rule; both give 10.

Power rule: for every positive integer n, d/dx (xⁿ) = nxⁿ⁻¹. It follows from expanding (x + h)ⁿ by the binomial theorem, or by induction using the product rule.

The power rule holds for all real powers too, as shown in later work. For instance d/dx (x³) = 3x², which is 12 at x = 2.