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MathematicsA Level & IB

What is product rule?

The product rule differentiates the product of two functions: differentiate the first times the second, plus the first times the derivative of the second.

Product rule explained

Set uu and vv clearly, write out dudx\frac{du}{dx} and dvdx\frac{dv}{dx}, then apply the formula. Neat labelling almost always avoids sign errors.
Many exam questions combine the product rule with the chain rule, for example differentiating x2e3xx^{2}e^{3x}.

Key formula

ddx(uv)=udvdx+vdudx\dfrac{d}{dx}(uv) = u\dfrac{dv}{dx} + v\dfrac{du}{dx}

Worked example

For y=x2sin⁡xy = x^{2}\sin x: dydx=x2cos⁡x+2xsin⁡x\frac{dy}{dx} = x^{2}\cos x + 2x\sin x.

Examiner tip

Factorise your answer where possible — later parts often ask for stationary points, which are far easier from a factorised derivative.

Product rule: common questions

Is there a quotient rule too?
Yes, and it is derived from the product rule. You can also rewrite a quotient as a product with a negative power and use the chain rule instead.
Does the product rule work for three functions?
Yes — apply it twice, treating two of the factors as a single product first.

Related terms

Chain rule

The chain rule differentiates a composite function — a function inside another function — by multiplying the derivatives of the outer and inner parts.

Differentiation

Differentiation is the process of finding the rate at which one quantity changes with respect to another — the gradient of a curve at a point.

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