Here we are asked to find the derivative of
the function F of X, where F of X is defined
as the integral from 3 to X of sine of 7T
plus 11 cubed DT.
So the first thing you notice is that F of
X is an accumulation function.
This means that it's finding the area under
a curve from a number to X.
Since it's an accumulation function, and we
want to take its derivative, we're taking
the derivative of an integral from a number
to X, we can use the first fundamental theorem
of calculus.
The first fundamental theorem of calculus
tells us that when we take the derivative
in terms of X, of an integral from a constant
up to X, we just get back the original function
on the inside but the variable changes from
T to X.
So this answer is that F prime of X is sine
of 7X plus 11 cubed.
Note that we don't change anything about the
inside of the function, except change the
T to X.
Let's try this again.
Here it says to find the derivative of G of
X equals the integral from X to 3 of 7 over
T plus 1 DT.
Here, we almost have an accumulation function.
We can almost use the first fundamental theorem
of calculus.
The problem is that the X is on the bottom
and the 3 is on the top.
Thankfully, we have a rule for definite integrals
that allows us to flip the boundaries, interchanging
A and B, but to do that we have to put a negative
sign out in front.
So we rewrite G of X as negative the integral
from 3 to X of 7 over T plus 1 DT.
Now we can take the derivative using the first
fundamental theorem of calculus.
We get G prime of X is equal to negative the
inside function, changing the T's to X's.
We get 7 over X plus 1.
Now let's try one that's a little bit trickier.
Here we're asked to find the derivative of
H of X where H of X is the integral from 0
to X cubed of 17 cosine of X DX.
Here, this one is also not quite an accumulation
function because of the X cubed.
And in fact, the way we want to think of it
is a composition.
We have some accumulation function F of X,
which is the integral from 0 to X of 17 cosine
of X DX.
And now H of X is equal to F of X cubed.
If we want to take the derivative of H of
X, we want to take the derivative of F of
X cubed.
And to do that, we need to use the chain rule.
The chain rule says we take the derivative
of F, we plug in X cubed, and then we multiply
that by the derivative of X cubed, which would
be 3 X squared.
Then the question is, what is the derivative
of F?
Well, F here really is an accumulation function.
It's an integral from 0, a constant, to X
and we want to take its derivative in terms
of X.
We can say that F prime of X really is, using
the first fundamental theorem of calculus,
17, ugh oh, 17 cosine of X.
Why did I say ugh oh?
Because in this problem, I've written it poorly.
I'm using X to mean two different things.
I'm using X to be the boundary at the top
of the integral and also the variable of integration.
That's bad practice, so let's change these
ones to T. That way, when we use the first
fundamental theorem, we change the T's to
X's.
And we don't have to use T for the inside,
this works just as well if you want to use
any other letter that you want.
How about, let's use, Y.
Y is a good letter.
But in any case, you use the first fundamental
theorem of calculus if the boundary letter
is the same as the function that you want
to take the derivative of.
If we're taking the derivative in terms of
X, and you have an integral from a constant
to X, then we want to use the first fundamental
theorem of calculus.
Have we finished this problem?
Not quite.
We found F prime of X, but now we need to
put that together.
What we really wanted was H prime of X.
So we have H prime of X is supposed to be
F prime of X cubed, so F prime of X is 17
cosine of X, so F prime of X cubed is 17 cosine
of X cubed, times the derivative of X cubed
which is 3 X squared.
Here we've done the first fundamental theorem
of calculus combined with the chain rule.
