Rules of differentiation

First principle always works but is slow. These rules give derivatives in one or two lines.

Sums and constants. (u ± v)′ = u′ ± v′, and (cu)′ = c u′, so a polynomial is differentiated term by term.

The power rule: d/dx xⁿ = nxⁿ⁻¹: bring the power down and reduce it by 1.

• d/dx (4x³ – 2x² + 7x – 9) = 12x² – 4x + 7.
• Roots and fractions are powers too: √x = x^(1/2) gives 1/(2√x), and 1/x³ = x⁻³ gives –3/x⁴.
• The slope of y = x³ – 3x at x = 2 is 3(4) – 3 = 9.

Product rule: (uv)′ = u′v + uv′. Keep one, differentiate the other, then swap and add. (It is not u′v′.)

• x² sin x: 2x sin x + x² cos x.
• (x + 2)(x² – 1): 1 × (x² – 1) + (x + 2) × 2x = 3x² + 4x – 1. Check: expanding first gives x³ + 2x² – x – 2, and its derivative is the same.

Quotient rule: (u/v)′ = (u′v – uv′)/v². Bottom times derivative of top, minus top times derivative of bottom, all over the bottom squared.

• (2x + 3)/(x – 1): [2(x – 1) – (2x + 3) × 1]/(x – 1)² = –5/(x – 1)².

The six trig derivatives:

A pattern to remember them: the ones starting with co (cos, cot, cosec) have a minus sign.

These follow from the rules. tan x = sin x/cos x, so by the quotient rule (cos x · cos x + sin x · sin x)/cos²x = 1/cos²x = sec²x.

• x cos x: cos x – x sin x.
• cos²x = cos x · cos x: –sin x cos x – cos x sin x = –2 sin x cos x = –sin 2x.

Powers of a bracket, like (2x + 1)³, can be done as a product of brackets. In Class 12 the chain rule makes it quick: differentiate the outside, then multiply by the derivative of the inside: 3(2x + 1)² × 2 = 6(2x + 1)².