Derivative Rules

The derivative f'(x) gives the instantaneous rate of change (the slope of the tangent). After limits, AP Calculus AB is largely about applying four differentiation rules quickly and correctly.

The four core rules

• Power rule: d/dx [xⁿ] = n·xⁿ⁻¹
• Product rule: (uv)' = u'v + uv'
• Quotient rule: (u/v)' = (u'v − uv') / v²
• Chain rule: d/dx [f(g(x))] = f'(g(x)) · g'(x)

Worked example (chain rule)

Differentiate y = (3x² + 1)⁴.

Outer power rule × inner derivative:

y' = 4(3x² + 1)³ · (6x) = 24x(3x² + 1)³

Common derivatives to memorise

d/dx[sin x] = cos x, d/dx[cos x] = −sin x, d/dx[eˣ] = eˣ, d/dx[ln x] = 1/x.

AP strategy

In the multiple-choice section, speed matters — recognise which rule applies at a glance (a product and a composition often need product and chain together). In the free-response section, show each rule step clearly: AP graders award method marks even if the final arithmetic slips. Nested functions almost always mean the chain rule — the most tested and most forgotten.