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
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.