Use the product rule when two functions are multiplied together.

Examples:

• \(x^2 \sin x\)
• \((3x-1)e^x\)
• \((x^2+1)\ln x\)

If \[ y=u\times v, \] then

Differentiate the first function and keep the second the same, then add:

first \(\times\) derivative of second
+
second \(\times\) derivative of first

Differentiate

\[ y=x^2\sin x \]

Let

\[ u=x^2, \qquad v=\sin x \]

Then

\[ \frac{du}{dx}=2x, \qquad \frac{dv}{dx}=\cos x \]

Using the product rule:

\[ \frac{dy}{dx}=x^2\cos x+\sin x(2x) \]

Differentiate

\[ y=(3x-1)e^x \]

Let

\[ u=3x-1, \qquad v=e^x \]

Then

\[ \frac{du}{dx}=3, \qquad \frac{dv}{dx}=e^x \]

So

\[ \frac{dy}{dx}=(3x-1)e^x+e^x(3) \]

\[ \frac{dy}{dx}=e^x(3x+2) \]

Differentiate

\[ y=(x^2+1)\ln x \]

Let

\[ u=x^2+1, \qquad v=\ln x \]

Then

\[ \frac{du}{dx}=2x, \qquad \frac{dv}{dx}=\frac{1}{x} \]

So

\[ \frac{dy}{dx}=(x^2+1)\cdot\frac{1}{x}+\ln x\cdot 2x \]

If there are more than two factors, it is often easier to group two of them first.

Example:

\[ y=x^2e^x\sin x \]

You can treat \(x^2\) as one function and \(e^x\sin x\) as the other, then use the product rule again if needed.

• Do not differentiate both functions at the same time and multiply them only.
• Do not forget the plus sign in the middle.
• Keep one function unchanged each time.
• Simplify only after applying the rule correctly.