Product rule
(uv)' = u'v + uv'. For products of two differentiable functions.
Y13 differentiation. Apply when the function is a product of two pieces. e.g. y = x sin x has u = x, v = sin x. y' = sin x + x cos x.
Worked examples
y = x² ex: y' = 2x ex + x² ex.
y = (3x+1) ln x: y' = 3 ln x + (3x+1)/x.
y = sin x cos x: y' = cos²x − sin²x = cos 2x.
Frequently asked questions
Memorise as ‘first times derivative of second plus derivative of first times second’.
u'v + uv'. Some prefer u dv + v du notation.
More than two factors?
Apply repeatedly; or take logs. d/dx(uvw) = u'vw + uv'w + uvw'.