Video: Finding the Third Derivative of Trigonometric Functions Using the Product Rule

Video Transcript

Find the third derivative of the function 𝑦 is equal to 44𝑥 times the sin of two 𝑥.

The question is asking us to find the third derivative of our function. And since we want to find the third derivative of our function in terms of 𝑥, we need to differentiate this three times with respect to 𝑥. Let’s start by finding the first derivative of 𝑦 with respect to 𝑥. That’s the derivative of 44𝑥 times the sin of two 𝑥 with respect to 𝑥. And we can see this is the derivative of the product of two functions. It’s the product of 44𝑥 and the sin of two 𝑥. So we’ll differentiate this by using the product rule.

The product rule tells us the derivative of the product of two functions 𝑢 and 𝑣 with respect to 𝑥 is equal to 𝑣 times d𝑢 by d𝑥 plus 𝑢 times d𝑣 by d𝑥. So to evaluate our derivative, we’ll set 𝑢 to be 44𝑥 and 𝑣 to be the sin of two 𝑥. To apply the product rule, we need to find expressions for d𝑢 by d𝑥 and d𝑣 by d𝑥. Let’s start with d𝑢 by d𝑥. That’s the derivative of 44𝑥 with respect to 𝑥. And this is a linear function in terms of 𝑥. So its derivative is just the coefficient of 𝑥, which in this case is 44.

Let’s now find an expression for d𝑣 by d𝑥. That’s the derivative of the sin of two 𝑥 with respect to 𝑥. And we know how to differentiate this using one of our standard trigonometric derivative results. For any real constant 𝑎, the derivative of the sin of 𝑎𝑥 with respect to 𝑥 is equal to 𝑎 times the cos of 𝑎𝑥. Applying this, we get d𝑣 by d𝑥 is equal to two times the cos of two 𝑥. We’re now ready to apply the product rule to find the d𝑦 by d𝑥. It’s equal to 𝑣 times d𝑢 by d𝑥 plus 𝑢 times d𝑣 by d𝑥.

Substituting in our expressions for 𝑢, 𝑣, d𝑢 by d𝑥, and d𝑣 by d𝑥, we get the sin of two 𝑥 times 44 plus 44𝑥 times two times the cos of two 𝑥. Then we just simplify and rearrange this expression. We get 44 sin of two 𝑥 plus 88𝑥 cos of two 𝑥. Remember, we want to find the third derivative of 𝑦 with respect to 𝑥 by differentiating our expression three times. So let’s now find our second derivative of 𝑦 with respect to 𝑥. That’s the derivative of our first derivative of 𝑥 with respect to 𝑥. We get the following expression. We see we can differentiate the first term by using our derivative rule for trigonometric functions.

However, the second term we need to differentiate is the product of two functions. So we’re going to need to use the product rule again. This time, we want 𝑢 to be 88𝑥 and we want 𝑣 to be the cos of two 𝑥. We need to find expressions for d𝑢 by d𝑥 and d𝑣 by d𝑥. Let’s start with d𝑢 by d𝑥. That’s the derivative of 88𝑥 with respect to 𝑥. And again, this is a linear function. So its derivative is the coefficient of 𝑥, which in this case is 88.

Let’s now find an expression for d𝑣 by d𝑥. That’s the derivative of the cos of two 𝑥 with respect to 𝑥. Again, we can evaluate this derivative by using one of our standard trigonometric derivative results. For any constant 𝑎, the derivative of the cos of 𝑎𝑥 with respect to 𝑥 is equal to negative 𝑎 times the sin of 𝑎𝑥. Applying this gives us d𝑣 by d𝑥 is equal to negative two times the sin of two 𝑥. We’re now ready to find an expression for the second derivative of 𝑦 with respect to 𝑥.

First, we’ll differentiate our first term, 44 times the sin of two 𝑥, to get 88 times the cos of two 𝑥. Next, we want to differentiate our second term by using the product rule. It’s equal to 𝑣 times d𝑢 by d𝑥 plus 𝑢 times d𝑣 by d𝑥. Substituting in our expressions for 𝑢, 𝑣, d𝑢 by d𝑥, and d𝑣 by d𝑥. We get that the second derivative of 𝑦 with respect to 𝑥 is equal to 88 times the cos of two 𝑥 plus the cos of two 𝑥 times 88 plus 88𝑥 times negative two times the sin of two 𝑥. And we can then simplify this expression to get 176 times the cos of two 𝑥 minus 176𝑥 sin two 𝑥.

The last thing we need to do is differentiate this expression to find our third derivative of 𝑦 with respect to 𝑥. So let’s clear some space so we can find our third derivative of 𝑦 with respect to 𝑥. That’s the derivative of 176 cos of two 𝑥 minus 176𝑥 sin of two 𝑥 with respect to 𝑥.

Again, we want to evaluate this derivative term by term. We can evaluate our first derivative by using our trigonometric derivative result. To evaluate our second derivative, we again notice it’s the product of two functions. So to evaluate the derivative of our second term, we’ll use the product rule. We’ll set 𝑢 to be 176𝑥 and 𝑣 to be the sin of two 𝑥. To use the product rule, we need expressions for d𝑢 by d𝑥 and d𝑣 by d𝑥. Let’s start with d𝑢 by d𝑥. That’s the derivative of 176𝑥 with respect to 𝑥. And again, this is a linear function. So its derivative is just the coefficient of 𝑥 which in this case is 176.

We now want to find an expression for d𝑣 by d𝑥. That’s the derivative of the sin of two 𝑥 with respect to 𝑥. And we can do this by using our other standard trigonometric derivative result. Applying this gives us d𝑣 by d𝑥 is equal to two times the cos of two 𝑥. We’re now ready to find an expression for d three 𝑦 by d𝑥 cubed. First, let’s differentiate the first term. Differentiating this gives us negative two times 176 times the sin of two 𝑥, which is negative 352 sin of two 𝑥. We then want to subtract the derivative of our second term, which by the product rule is 𝑣 times d𝑢 by d𝑥 plus 𝑢 times d𝑣 by d𝑥.

Substituting in our expressions for 𝑢, 𝑣, d𝑢 by d𝑥, and d𝑣 by d𝑥, we have the third derivative of 𝑦 with respect to 𝑥 is equal to negative 352 sin of two 𝑥 minus the sin of two 𝑥 times 176 plus 176𝑥 times two times the cos of two 𝑥. Then we just simplify and rearrange this expression. We get negative 528 sin of two 𝑥 minus 352𝑥 times the cos of two 𝑥. And finally, we’ll reorder our terms so that our powers of 𝑥 are descending.

Therefore, we’ve shown the function 𝑦 is equal to 44𝑥 times the sin of two 𝑥 has a third derivative equal to negative 352𝑥 cos of two 𝑥 minus 528 sin two 𝑥.