Product rule

The product rule is used to differentiate the product of two functions. If y = uv where u and v are both functions of x, then dy/dx = u(dv/dx) + v(du/dx), often written as (uv)' = uv' + vu'. This rule is essential when you cannot simply expand the product. For example, y = x²sin(x) cannot be expanded, so we use the product rule with u = x² and v = sin(x). To apply the product rule: identify u and v, find du/dx and dv/dx separately, then combine using the formula. For y = x²sin(x): du/dx = 2x, dv/dx = cos(x), so dy/dx = x²cos(x) + 2x·sin(x). The product rule can be extended to three or more functions: (uvw)' = u'vw + uv'w + uvw'. However, it is often easier to apply the rule twice: first differentiate uv as a single function multiplied by w. Common mistakes include forgetting one of the terms, or incorrectly differentiating u or v. Always clearly identify your u, v, du/dx, and dv/dx before combining. Check your answer by considering whether the result has the expected form.