2.8 The Product Rule

The rule

For a product of two functions: d/dx[f·g] = f'·g + f·g'. It produces two terms — the derivative of a product is NOT the product of the derivatives.

Example

d/dx[x²·sin x] = (2x)(sin x) + (x²)(cos x) = 2x·sin x + x²·cos x.

Key terms

Product rule

d/dx[f·g] = f'g + fg'.

When to use

Two functions multiplied together.

Not this

The derivative of a product is NOT f'·g'.

Two terms

One for each factor's derivative.

Example

d/dx[x²·sin x] = 2x·sin x + x²·cos x.

Try an AP-style question

d/dx (x² sin x) = ?

1. 2x sin x + x² cos x
2. 2x cos x
3. x² cos x
4. 2x sin x − x² cos x