How Do You Use the Product Rule?

Use the product rule when a function is formed by multiplying two functions of the same variable. Differentiate each factor separately, then use $\frac{d}{dx}[u(x)v(x)]=u'(x)v(x)+u(x)v'(x)$.

The Method

Suppose

$$y=u(x)v(x).$$

Then

$$\frac{dy}{dx}=u'(x)v(x)+u(x)v'(x).$$

This means: differentiate the first function and leave the second unchanged, then leave the first unchanged and differentiate the second. Add the two resulting terms.

Follow these steps:

1. Identify the two multiplied functions, $u(x)$ and $v(x)$.
2. Calculate (u'(x)) and (v'(x)) separately.
3. Substitute into $u'v+uv'$, keeping both terms.

The rule works because both factors change when $x$ changes. The first term measures the change caused by $u$, while the second measures the change caused by $v$. Therefore, differentiating only one factor cannot represent the total rate of change.

Choosing the factors well also makes the method clearer. For example, for $y=x\sin x$, take $u=x$ and $v=\sin x$. Then $u^{\mathrm{\prime }}$ and , so

$$\frac{dy}{dx}=\sin x+x\cos x.$$

Here, the first product-rule term differentiates $x$, while the second differentiates $\sin x$. Neither term may be omitted, even though one derivative is simply $1$ in this straightforward example.

Worked Example

Differentiate

$$y=(x^2+3)e^x.$$

Let $u(x)=x^2+3$ and $v(x)=e^x$. Therefore,

$$u'(x)=2x \quad \text{and} \quad v'(x)=e^x.$$

Applying the product rule gives

$$\frac{dy}{dx}=(2x)e^x+(x^2+3)e^x.$$

Factorizing the common factor $e^x$ produces

$$\frac{dy}{dx}=e^x(x^2+2x+3).$$

The common misconception is to write (u'(x)v'(x)). This is incorrect: the derivative of a product is generally not the product of the derivatives. Both (u'v) and (uv') are required. Another error is differentiating one factor but accidentally changing the other factor in the same term.

Exam Technique

This is IB Mathematics: Analysis and Approaches, SL 5.6, and it may appear on Paper 1 or Paper 2. Examiners expect the method to be visible, so write the product-rule substitution before simplifying. A correct unsupported answer may not receive every method mark. Also check whether either factor is composite and requires the chain rule; IB questions frequently combine the product and chain rules.