Differentiation Rules

The Product Rule

Find the derivative of a product of two functions using the product rule.

Example

Differentiate both factors and assemble u prime v plus u v prime.

highlighted = computed this step

Step 1 — Set up

Start with a product of two functions.

f(x)=x2sin⁡xf(x)= x^{2}\sin x

Step 2 — Name the factors

Name the first factor u and second factor v.

u=x2v=sin⁡xu= \hlmath{x^{2}} \quad v= \hlmath{\sin x}

Step 3 — Differentiate factors

Differentiate u and v separately.

u′=2xv′=cos⁡xu'= \hlmath{2x} \quad v'= \hlmath{\cos x}

Step 4 — Product rule

Apply the product rule.

(uv)′=u′v+uv′(uv)'=u'v+uv'

Step 5 — Assemble result

Assemble u prime v plus u v prime.

f′(x)=2xsin⁡x+x2cos⁡xf'(x)= \hlmath{2x\sin x+x^{2}\cos x}
product-rule If f(x) = u(x)·v(x), then f'(x) = u'v + uv'. Differentiate the first factor times the second, plus the first times the derivative of the second.