Lesson Video: The Product Rule Mathematics

Video Transcript

In this video, we’ll learn how to apply the product rule to find the derivative of a function which is the product of two or more other functions. We’ll begin by learning what the product rule actually is before applying the rule to help us find the derivative of a number of functions, including those which are themselves are product of at least three different functions. We’ll also briefly consider the application of the product rule for calculating the coordinates of the critical points or stationary points on a graph.

Consider the example of 𝑥 squared plus three 𝑥 plus one times 𝑥 minus four 𝑥 to the power of five.

Let’s think how we could find the derivative of this expression with respect to 𝑥. In the past, we might have looked to distribute this parentheses before applying the normal rules for differentiation to each of the resultant terms. In fact though, this expression is the product of two separate functions. The first is the function 𝑥 squared plus three 𝑥 plus one. The second is the function 𝑥 minus four 𝑥 to the power of five.

And there’s a rule that will allow us to differentiate any expression which is the product of two or more functions. Funnily enough, this rule is called the product rule. Now, the proof of the product rule is somewhat long-winded and that’s outside of the constraints of this video. Instead, we’ll simply state the product rule and consider its application to calculus.

The product rule states that the derivative of the product of functions 𝑓 and 𝑔 is equal to 𝑓 multiplied by the derivative of 𝑔 plus the derivative of 𝑓 multiplied by 𝑔. And this is sometimes alternatively written as the derivative of 𝑢𝑣 with respect to 𝑥 is 𝑢 times d𝑣 by d𝑥 plus 𝑣 times d𝑢 by d𝑥, where here 𝑢 and 𝑣 are functions in 𝑥. It’s very much a matter of personal preference as to which you decide to use. But this rule does need to be memorized. We’ll begin by considering a simple example of the application of this rule.

Find the first derivative of the function 𝑦 equals three 𝑥 to the power of five plus seven multiplied by seven minus three 𝑥 to the power of five.

Here, we have an equation which is the product of two functions. Since this is the case, we can find the derivative of this function by applying the product rule. Remember this rule says that the derivative of the product of 𝑓 and 𝑔 is equal to 𝑓 times the derivative of 𝑔 plus the derivative of 𝑓 times 𝑔. It’s always sensible to write down what we actually know about the expression we’re looking to differentiate. We can say that it’s the product of two functions. Let’s call the first one 𝑓 of 𝑥 three 𝑥 to the power of five plus seven. And we’ll call the second 𝑔 of 𝑥. It’s seven minus three 𝑥 to the power of five. We’re going to need to find the first derivative of each of these functions.

Remember the derivative of a simple algebraic expressions such as 𝑎𝑥 to the power of 𝑛, where 𝑎 and 𝑛 are real constants, is 𝑛 times 𝑎𝑥 to the power of 𝑛 minus one. We multiply by the exponent and then reduce that exponent by one. And what this does actually mean is that the derivative of a constant is zero since a constant — let’s say the number one — is actually one 𝑥 to the power of zero. When we multiply by the exponent, we actually just end up with an answer of zero. This means the derivative of 𝑓 of 𝑥 is five times three 𝑥 to the power of four plus zero which is simply 15𝑥 to the power of four. Similarly, the derivative of 𝑔 of 𝑥 is zero minus five times three 𝑥 to the power of four which is negative 15𝑥 to the power of four.

Let’s substitute what we know into the formula for the product rule. d𝑦 by d𝑥 is equal to three 𝑥 to the power of five plus seven times negative 15𝑥 to the power of four plus 15𝑥 to the power of four times seven minus three 𝑥 to the power of five. We then distribute these parentheses by multiplying each term by the term outside. And we get negative 45𝑥 to the power of nine minus 105𝑥 to the power of four plus 105𝑥 to the power of four minus 45𝑥 to the power of nine. Negative 105𝑥 to the power of four plus 105𝑥 to the power of four is zero. And we can see that the first derivative of our function is negative 90𝑥 to the power of nine. It’s also useful to know that we can apply this rule to find the derivative at a given point.

Let’s consider an example of this.

Find the first derivative of 𝑓 of 𝑥 equals nine 𝑥 squared minus 𝑥 minus seven times seven 𝑥 squared minus eight 𝑥 minus seven at negative one, 24.

Here, we have an equation which is the product of two functions. Since this is the case, we can find the derivative of this function by applying the product rule. Remember that says that we can find the derivative of the product of 𝑓 and 𝑔 by finding 𝑓 times the derivative of 𝑔 plus the derivative of 𝑓 times 𝑔. Now, actually our equation is 𝑓 of 𝑥. So we’re going to split our two functions into 𝑔 of 𝑥 and ℎ of 𝑥. We can say that 𝑔 of 𝑥 is equal to nine 𝑥 squared minus 𝑥 minus seven and ℎ of 𝑥 is equal to seven 𝑥 squared minus eight 𝑥 minus seven. And then, I’ve changed the product rule. This time, it says the derivative of 𝑔 times ℎ is 𝑔 times the derivative of ℎ plus the derivative of 𝑔 times ℎ. Let’s find the derivative of 𝑔.

The derivative of nine 𝑥 squared is 18𝑥 and the derivative of negative 𝑥 is negative one. So the derivative of 𝑔 of 𝑥 is 18 𝑥 minus one. Similarly, the derivative of ℎ of 𝑥 is 14𝑥 minus eight. Let’s substitute what we know into the formula for the product rule. The derivative of 𝑓 with respect to 𝑥 is therefore nine 𝑥 squared minus 𝑥 minus seven times 14𝑥 minus eight plus seven 𝑥 squared minus eight 𝑥 minus seven times 18𝑥 minus one. We distribute the parentheses carefully and collect like terms. And we see that the derivative of 𝑓 of 𝑥 is 252𝑥 cubed minus 237𝑥 squared minus 208𝑥 plus 63.

But we aren’t quite finished. We’ve been asked to find the derivative at the point negative one, 24. This is the point in our Cartesian plane, where 𝑥 is equal to negative one and 𝑦 is equal to 24. So we can substitute 𝑥 is equal to negative one into our expression for the derivative. That’s 252 times negative one cubed minus 237 times negative one squared minus 208 times negative one plus 63, which is negative 218. In fact, it’s useful to remember that this actually tells us the gradient of the tangent to the curve at the point negative one, 24.

In our next example, we’ll consider how to apply the product rule to differentiate an expression which is the product of more than two functions.

The product rule says that the derivative of 𝑓𝑔 is equal to the derivative of 𝑓 times 𝑔 plus 𝑓 times the derivative of 𝑔. Use this to derive a formula for the derivative of 𝑓 times 𝑔 times ℎ.

In this question, we’ve been given the product rule and asked to use it to find a formula for the derivative of the product of three functions. These are 𝑓, 𝑔, and ℎ. We’re going to begin by splitting 𝑓 times 𝑔 times ℎ up. We’re going to write it as 𝑓𝑔 times ℎ. Remember since multiplication is commutative, we could have alternatively written it as 𝑓 times 𝑔ℎ and we will get the same answer either way. So we can say that the derivative of 𝑓𝑔ℎ is equal to the derivative of 𝑓𝑔 times ℎ.

And we’re now going to apply the product rule. We can see that this is equal to the derivative of 𝑓𝑔 times ℎ plus 𝑓𝑔 times the derivative of ℎ. And now, we spot that the first term we have is the derivative of 𝑓𝑔. We know though by the definition of the product rule that this is the same as the derivative of 𝑓 times 𝑔 plus 𝑓 times the derivative of 𝑔. So we replace this in our formula. And we’re going to distribute these parentheses.

When we do, we see that the formula for the derivative of 𝑓𝑔ℎ is the derivative of 𝑓 times 𝑔 times ℎ plus 𝑓 times the derivative of 𝑔 times ℎ plus 𝑓 times 𝑔 times the derivative of ℎ. You might also like to see if you can apply this idea to help you find a formula for the derivative of the product of four functions, say 𝑓𝑔ℎ𝑖.

Next, we’re going to look at how this example can help us to find the derivative of an expression which is the product of three functions.

Find the first derivative of 𝑓 of 𝑥 equals 𝑥 to the power of eight plus four times three 𝑥 root 𝑥 minus seven times three 𝑥 root 𝑥 plus seven at 𝑥 equals negative one.

To answer this question, we have two options. We could use the product rule twice or we could recall the definition of the derivative of the product of three functions. The derivative of 𝑓 times 𝑔 times ℎ is the derivative of 𝑓 times 𝑔 times ℎ plus 𝑓 times the derivative of 𝑔 times ℎ plus 𝑓 times 𝑔 times the derivative of ℎ. Now since our function is actually 𝑓 of 𝑥, I’ve changed the functions in this formula to be 𝑢 𝑣 and 𝑤. So let’s work out what the functions 𝑢, 𝑣, and 𝑤 actually are. We can say that 𝑢 of 𝑥 is equal to 𝑥 to the power of eight plus four. 𝑣 of 𝑥 is equal to three 𝑥 root 𝑥 minus seven. And 𝑤 of 𝑥 is equal to three 𝑥 root 𝑥 plus seven.

We need to differentiate each of these functions with respect to 𝑥 as per the formula for the product rule with three functions. The derivative of 𝑢 is fairly straightforward. It’s just eight 𝑥 to the power of seven. But what about 𝑣 and 𝑤? Well, we could use the product rule. But actually, we can simply rewrite each of these expressions. We know that the square root of 𝑥 is the same as 𝑥 to power of one-half. And the laws of exponents tell us we can simplify this expression by adding the powers.

And when we do, we see that 𝑣 of 𝑥 can be written as three 𝑥 to the power of three over two minus seven and 𝑤 of 𝑥 is three 𝑥 to the power of three over two plus seven. And this means the derivative of the 𝑣 is three over two times three 𝑥 to the power of one-half or nine over two 𝑥 to the power of one-half. And actually, the derivative of 𝑤 is the same.

We now have everything we need to substitute this into our formula for the product rule. Now, at this point, you might be tempted to jump straight into substituting 𝑥 is equal to negative one into the derivative. However, we have some roots here and that might cause issues. Instead, we carefully distribute each set of parentheses and simplify fully. And when we do, we see that the derivative of 𝑓 of 𝑥 is 99𝑥 to the power of 10 minus 392𝑥 to the power of seven plus 108𝑥 to the power of two.

And we can now evaluate this at 𝑥 is equal to negative one. It’s 99 times negative one to the power of 10 minus 392 times negative one to the power of seven plus 108 times negative one squared which is equal to 599.

In our final example, we’ll consider how to use the product rule to solve problems involving critical points.

Find the coordinates of the critical points on the curve with equation 𝑦 equals 𝑥 over 16 plus 𝑥 squared.

The critical points of a curve are found when the derivative is made equal to zero. We’ll begin then by differentiating our equation 𝑦 in terms of 𝑥. We can begin by rewriting our equation as 𝑥 times 16 plus 𝑥 squared to the power of negative one. And we can now differentiate this using the product rule. We’ll let our first function — I’ve called that 𝑢 — be equal to 𝑥. And we’ll say our second function is 16 plus 𝑥 squared to the power of negative one. It’s quite straightforward to find the derivative of 𝑢. We just get one. But we’re going to need to use the chain rule to differentiate 16 plus 𝑥 squared to the power of negative one.

We’re going to let 𝑡 be equal to 16 plus 𝑥 squared. Then, differentiating 𝑡 with respect to 𝑥 gives us two 𝑥. And we can now say that 𝑣 is equal to 𝑡 to the power of negative one. We need to differentiate 𝑣 with respect to 𝑡. And when we do, we see that it’s negative 𝑡 to the power of negative two. The derivative of our function 𝑣 with respect to 𝑥 is equal to d𝑣 by d𝑡 times d𝑡 by d𝑥. At the moment, that’s negative 𝑡 to the power of negative two times two 𝑥. We can now replace 𝑡 with 16 plus 𝑥 squared. And we’ve differentiated 𝑣 with respect to 𝑥.

We can now clear some space and apply the product rule. It’s 𝑢 multiplied by the derivative of 𝑣 plus 𝑣 multiplied by the derivative of 𝑢. We then rewrite it slightly. We say that the derivative of 𝑦 with respect to 𝑥 is negative two 𝑥 squared over 16 plus 𝑥 squared all squared plus one over 16 plus 𝑥 squared. We’re going to simplify this expression by multiplying the denominator and the numerator of our second fraction by 16 plus 𝑥 squared. And we see that we now have negative two 𝑥 squared over 16 plus 𝑥 squared squared plus 16 plus 𝑥 squared over 16 plus 𝑥 squared squared.

By adding the numerators, we see we’ve successfully found the derivative. It’s negative 𝑥 squared plus 16 over 16 plus 𝑥 squared all squared. Now earlier, we said that the coordinates of the critical points could be found by letting the derivative be equal to zero. So we’re going to say our fraction negative 𝑥 squared plus 16 over 16 plus 𝑥 squared squared is equal to zero. And then, we consider what needs to be true for this to be the case. Well, it doesn’t actually matter what the denominator of our fraction is. If the numerator of our fraction is zero, then the entire fraction must be equal to zero.

So we say that negative 𝑥 squared plus 16 is equal to zero and we solve this. We add 𝑥 squared to both sides which gives us 𝑥 squared equals 16. And then, we find the square root of both sides of the equation, remembering to take both the positive and negative square roots of 16. And we end up with two values for 𝑥: positive and negative four. So the critical points of our curve occur when 𝑥 is equal to plus or minus four. We’re going to need to substitute this back into the original equation to find the coordinates of the critical points.

When 𝑥 is equal to four, 𝑦 is four over 16 plus four squared which is one-eighth. And when 𝑥 is equal to negative four, 𝑦 is negative four over 16 plus negative four squared which is negative one-eighth. So the coordinates of the critical points on the curve with equation 𝑦 equals 𝑥 over 16 plus 𝑥 squared are four, one-eighth and negative four, negative one-eighth.

In this video, we’ve learned that we can apply the product rule to find the derivative of the product of two functions. We’ve learned that for two functions 𝑓 and 𝑔, the derivative of their product is 𝑓 times the derivative of 𝑔 plus the derivative of 𝑓 times 𝑔. And finally, we’ve seen that whilst we can apply the product rule in succession, we can also find the product of three functions by using the given formula.