 # The Product Rule and the Quotient Rule

## The Product Rule and the Quotient Rule

#### The Product Rule

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 = xe3x 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 = xe3x, find the turning point and sketch the graph.

u = x

v = e3x Therefore: This means there is a stationary point when x = -1/3 (e3x ≠ 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

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: 