The product rule is used when differentiating two functions that are being multiplied together. In some cases it will be possible to simply multiply them out.

Example:

Differentiate y = x²(x² + 2x − 3).

Here y = x⁴ + 2x³ − 3x² and so:

However functions like y = 2x(x² + 1)⁵ and y = xe^3x are either more difficult or impossible to expand and so we need a new technique.

The product rule states that for two functions, u and v. If y = uv, then:

For our example:

y = 2x(x² − 1)⁵

u = 2x

v = (x² − 1)⁵

Therefore:

After factorising:

Note: After using the product rule you will normally be able to factorise the derivative and then you can find the stationary points.

For our second example:

y = xe^3x, find the turning point and sketch the graph.

u = x

v = e^3x

Therefore:

This means there is a stationary point when x = -1/3 (e^3x ≠ 0).

Also, when x = -1/3, y = -e^-1/3 = -0.123 (3sf).

By using the second derivative, which we find by using the product rule again, we can determine whether this is a maximum or a minimum.

when x = -1/3

Therefore there is a minimum at (-1/3, -0.123)

To sketch the graph we know that:

1. When x = 0, y = 0
2. There is a minimum at (-1/3, -0.123)
3. As x → ∞, y → ∞
4. As x → −∞, y → 0 (and is negative)

Therefore the graph looks like this:

The quotient rule is actually the product rule in disguise and is used when differentiating a fraction.

The quotient rule states that for two functions, u and v,

(See if you can use the product rule and the chain rule on y = uv^-1 to derive this formula.)

Example:

Differentiate

Solution: