Product Rule

The product rule tells us the derivative of two functions f and g that are multiplied together:

(fg)’ = fg’ + gf’

(The little mark ’ means "derivative of".)

Example: What is the derivative of cos(x)sin(x) ?

We have two functions cos(x) and sin(x) multiplied together, so let's use the Product Rule:

(fg)’ = f g’ + f’ g

Which in our case becomes:

(cos(x)sin(x))’ = cos(x) sin(x)’ + cos(x)’ sin(x)

We know (from Derivative Rules) that:

sin(x)’ = cos(x)
cos(x)’ = −sin(x)

So we can substitute:

(cos(x)sin(x))’ = cos(x) cos(x) + −sin(x) sin(x)

Which simplifies to:

(cos(x)sin(x))’ = cos²(x) − sin²(x)

Answer: the derivative of cos(x)sin(x) = cos²(x) − sin²(x)

Why Does It Work?

When we multiply two functions f(x) and g(x) the result is the area fg:

product rule

The derivative is the rate of change, and when x changes a little then both f and g will also change a little (by Δf and Δg). In this example they both increase making the area bigger.

How much bigger?

Increase in area = Δ(fg) = fΔg + ΔfΔg + gΔf

As the change in x heads towards zero, the "ΔfΔg" term also heads to zero, and we get:

(fg)’ = fg’ + gf’

Alternative Notation

An alternative way of writing it (called Leibniz Notation) is:

ddx(uv) = udvdx + vdudx

Here is our example from before in Leibniz Notation:

Example: What is the derivative of cos(x)sin(x) ?

This:

ddx(uv) = udvdx + vdudx

Becomes this:

ddx(cos(x)sin(x)) = cos(x)d(sin(x))dx + sin(x)d(cos(x))dx

From Derivative Rules:

ddxsin(x) = cos(x)
ddxcos(x) = −sin(x)

So:

ddx(cos(x)sin(x)) = cos(x) cos(x) + −sin(x) sin(x)

Which simplifies to:

ddx(cos(x)sin(x)) = cos²(x) − sin²(x)

Three Functions

For three functions multiplied together we can use:

(fgh)’ = f’gh + fg’h + fgh’