Prove the product rule.

Primary tabs

Lesson Parent: 
Differentiation IV: Differentiation Formulas Everyone Must Know
Problem: 

What is the derivative of the product of $f(x)$ and $g(x)$ where both functions are differentiable? i.e. what is \[\frac{d}{dx}\left[f(x)\,g(x)\right]?\]

Answer: 

$f(x)\,g'(x)+f'(x)\,g(x)$

SAMPLE SOLUTION

All of our problems have fully worked out, logical solutions. We connect every solution to its underlying lesson so you can understand the concepts needed to solve any problem and reinforce your learning by practicing with similar problems. All of the formulas are available to our subscribers, but here is a sample of the problem solution you will see when you log in:



Solution: 

Let's define the function <Sign in to see all the formulas>. Now we want to find

<Sign in to see all the formulas>


Now let's add and subtract <Sign in to see all the formulas> in the numerator to get

<Sign in to see all the formulas>

It's important to note several facts that we are given. First, we know that

<Sign in to see all the formulas>


and that

<Sign in to see all the formulas>

because we are told that both $f(x)$ and $g(x)$ are differentiable.

Because $f(x)$ is differentiable we know that it is continuous. This means that

<Sign in to see all the formulas>


(Remember that in this problem $x$ is viewed as being a constant. This is why we use $x'$ which is "moving" toward $x$.)
Furthermore, we know that

<Sign in to see all the formulas>

Therefore,

<Sign in to see all the formulas>

And our last fact is that because $x$ is viewed as a constant we know that the limit of a constant is equal to the constant. Therefore,

<Sign in to see all the formulas>

Ok, we are almost done. Let's return our thoughts back to

<Sign in to see all the formulas>


From the addition and multiplication of known limits this limit will result in

<Sign in to see all the formulas>

Since multiplication is commutative we know that <Sign in to see all the formulas> and we have

<Sign in to see all the formulas>

Our proof is now complete. This is called the product rule and is an extremely useful formula to find derivatives of two functions that are multiplied together.