Derivative of a Product of Functions - At A Glance

Find the derivative of h(x) = x²e^x.

h is a product of the two functions f(x) = x² and g(x) = e^x.

Then

f ' (x) = 2x and g ' (x) = e^x,

therefore

While the answer

2xe^x + x²e^x

is correct, we factored out xe^x to find the answer

xe^x(2 + x)

because in later problems, it will be helpful to have the answer in this factored form.

While the product rule is usually written

(fg)' = f 'g + fg ',

the letters "f" and "g" are placeholders. To find the derivative of a product of two functions, find what each function multiplied by the derivative of the other is, and add the results.

If we avoid the letters f and g, the product rule says: to find the derivative of

((first function) × (second function)),

find

(derivative of first function) × second function

and add 

first function × (derivative of second function).

We only need to use the product rule when there are two expressions with x multiplied together.

What is the derivative of f(x) = sin x cos x?

The function f is a product of two functions of x:

f(x) = (sin x)(cos x).

We need to use the product rule to find the derivative of f. In order to use the product rule we need to know the derivatives of the two functions multiplied together:

(sin x)' = cos x

(cos x)' = -sin x.

Now we can apply the product rule:

What's the derivative of g(x) = x²sin x?

We multiply each function by the derivative of the other one, then add:

Find the derivative of f(x) = 4x².

While f(x) = 4 × x², we don't need to use the product rule because 4 doesn't have any x's in it. We'll use the rule for multiplication-by-a-constant instead:

What's the derivative of

f(x) = x²e^xsin x?

In order to use the product rule, we need to write our function as a product of two other functions. There are several ways to do this. We can write

f(x) = (x²e^x)(sin x)

or we can write

f(x) = (x²)(e^xsin x).

Use the first version,

f(x) = (x²e^x)(sin x).

In order to use the product rule we need to find the derivative of each factor.

(sin x)' = cos x.

For the other, we need to use the product rule again:

Now we need to remember to put everything back together to find the derivative of the original function:

We have several choices of what to do with the final answer. The answer

f ' (x) = (2xe^x + x²e^x)sin x + x²e^xcos x

is a reasonable way to write it, since we can look at this answer and see that it came from the product rule. Another reasonable thing to do is factor out common factors to find

f ' (x) = xe^x[(2 + x)sin x + xcos x],

which might be helpful if we wanted to do something with the derivative later. We could also multiply everything out to find

f ' (x) = 2xe^xsin x + x²e^xsin x + x²e^xcos x.

The product rule also allows us to find derivatives of more complicated products. To find the derivative of 3 or more functions multiplied together, we need to be careful with our work, so that we don't lose track of things.