Video: Evaluating an Expression Containing a Higher Order Derivative of a Trigonometric Function

Video Transcript

If 𝑦 equals sine five 𝑥, find 25 multiplied by the first derivative squared plus the second derivative squared.

So in order to find this problem, what we need to do is actually find the first and second derivatives. So first of all, we gonna start with the first derivative. And our first derivative is gonna be five cos five 𝑥. And we actually got that because, first of all, if you differentiate sine 𝑥 you get cosine 𝑥, but also we’ve use the chain rule to give us the five cos five 𝑥.

Okay, so now we found the first derivative. What we can do is actually move on and find the second derivative. And the second derivative is gonna be negative 25 sine five 𝑥. And again, first of all, this is because actually if you differentiate cosine 𝑥, you get negative sine 𝑥, but also again we use the chain rule. But what we’ll do is actually to demonstrate how we used it just to remind you how the chain rule works.

So with the chain rule, we say that if we actually have it in the form 𝑦 is equal to 𝑓 of 𝑢, then 𝑑𝑦 𝑑𝑥 is gonna be equal to 𝑑𝑦 𝑑𝑢 multiplied by 𝑑𝑢 𝑑𝑥. So let’s show how that works with our problem. Well if we had 𝑦 is equal to five cos five 𝑥, well then our 𝑦 was gonna be equal to five cos 𝑢 and our 𝑢 will equal to five 𝑥. So then if we differentiate five cos 𝑢, we’re gonna get negative five sine 𝑢, which gives us 𝑑𝑦 𝑑𝑢. And if we differentiate five 𝑥, we just get five. That’s 𝑑𝑢 𝑑𝑥.

So therefore, 𝑑𝑦 𝑑𝑥 is gonna be equal to negative five sine 𝑢 multiplied by five cause that’s our 𝑑𝑦 𝑑𝑢 multiplied by our 𝑑𝑢 𝑑𝑥. And then, if we actually simplify and substitute back in our 𝑢 equals five 𝑥, we get 𝑑𝑦 𝑑𝑥 is equal to negative 25 sine five 𝑥. So great! That’s what we had when we found out the question. Okay, brilliant! So we’ve got the first derivative and we have the second derivative. So now let’s actually put them back into our expression to find out what 25 multiplied by the first derivative squared plus the second derivative squared is going to be.

So therefore, what we’ve got is 25 multiplied by the first derivative squared plus the second derivative squared is equal to 25 multiplied by five cos five 𝑥 all squared, and that’s because that was our first derivative, plus negative 25 sine five 𝑥 all squared. That’s cause that was our second derivative. So therefore, we’re gonna get this is equal to 625 cos squared five 𝑥.

And we got 625 because we had five squared, because that was in the parentheses, which gave us 25. And 25 multiplied by 25 gives us 625. And then this is plus 625 sine squared five 𝑥. We got this because we actually had negative 25 all squared, which gave us positive 625. And then we’ve got sine squared five 𝑥. Well then, what we can actually do is take out 625 as a factor. So we do that. We got 625 multiplied by cos squared five 𝑥 plus sine squared five 𝑥.

And this is really important because this is now in a very useful form. Well, it’s useful because we know that cos squared 𝑥 plus sine squared 𝑥 is equal to one. So therefore, cos squared five 𝑥 plus sine squared five 𝑥 is also gonna be the same as one. So therefore, we’re gonna have 625 multiplied by one because that’s because we had the cos squared five 𝑥 plus sine squared five 𝑥 in our parentheses, which is gonna give us a final answer of 625.

So we know that if 𝑦 equals sine five 𝑥, then 25 multiplied by the first derivative squared plus the second derivative squared is gonna be equal to 625.