Video Transcript
Use the fundamental theorem of
calculus to find the derivative of the function ℎ of 𝑥 is equal to the definite
integral from four to the square root of 𝑥 of three 𝑧 squared all divided by 𝑧 to
the fourth power plus two with respect to 𝑧.
We’re given a function ℎ of 𝑥
which is defined as a definite integral where a function of 𝑥 is one of the limits
of integration. We need to find the derivative of
this with respect to 𝑥 by using the fundamental theorem of calculus. So let’s start by recalling the
fundamental theorem of calculus. In fact, we’ll only recall part of
this. We recall this tells us if
lowercase 𝑓 is a continuous function on a closed interval from 𝑎 to 𝑏 and capital
𝐹 of 𝑥 is the definite integral from 𝑎 to 𝑥 of lowercase 𝑓 of 𝑧 with respect
to 𝑧, then capital 𝐹 prime of 𝑥 will be equal to lowercase 𝑓 of 𝑥 for all
values of 𝑥 in the open interval from 𝑎 to 𝑏.
In other words, this gives us a
method of finding the derivative of an integral where 𝑥 is one of the limits of
integration. However, we can’t directly apply
this in this case. We can see the upper limit of
integration is not 𝑥. It’s a function of 𝑥. In fact, we can get around this by
setting the function 𝑢 of 𝑥 to be equal to the square root of 𝑥 and then applying
the chain rule. However, it’s also worth noting we
can do this in the general case and rewrite our definition for the fundamental
theorem of calculus.
Assuming that 𝑢 of 𝑥 is a
differentiable function, we get that 𝑓 is continuous on the closed interval from 𝑎
to 𝑏 and capital 𝐹 of 𝑥 is the integral from 𝑎 to 𝑢 of 𝑥 of 𝑓 of 𝑧 with
respect 𝑧. Then by using our standard
statement for the fundamental theorem of calculus and the chain rule, we get capital
𝐹 prime of 𝑥 is equal to lowercase 𝑓 evaluated at 𝑢 of 𝑥 times the derivative
of 𝑢 of 𝑥 with respect to 𝑥, so long as 𝑢 of 𝑥 in the open interval from 𝑎 to
𝑏. And we can apply this version
instead for our question. Our function 𝑢 of 𝑥 will be the
square root of 𝑥. Of course, we know this is a
differentiable function.
We see that the lower limit of
integration, four, will be our value of 𝑎. Capital 𝐹 of 𝑥 is the function
we’re differentiating. In this case, this is ℎ of 𝑥. And our integrand of three 𝑧
squared divided by 𝑧 to the fourth power plus two is our function lowercase 𝑓 of
𝑧. And remember, to use the
fundamental theorem of calculus, we still need to check where our integrand
lowercase 𝑓 of 𝑧 is continuous. In this case, our integrand is a
rational function. It’s the quotient of two
polynomials.
This means it’s continuous on this
entire domain. And we know the only time a
rational function won’t be defined is when its denominator is equal to zero. And if we solve the denominator, 𝑧
to the fourth power plus two, is equal to zero, we would find it has no real
solutions. Therefore, our integrand is
continuous on the entire set of real numbers. So in particular, it will be
continuous on any closed interval. Therefore, we can use this version
of the fundamental theorem of calculus. So by applying this version of the
fundamental theorem of calculus, which we got by using the chain rule, we get that ℎ
prime of 𝑥 is equal to 𝑓 evaluated at 𝑢 of 𝑥 times the derivative of 𝑢 of 𝑥
with respect to 𝑥.
The first thing we’ll do to this
expression is write the square root of 𝑥 instead of 𝑢 of 𝑥. This gives us 𝑓 evaluated at root
𝑥 times the derivative of root 𝑥 with respect to 𝑥. Remember, lowercase 𝑓 is our
integrand, so we need to substitute 𝑧 is equal to root 𝑥 into our integrand three
𝑧 squared divided by 𝑧 to the fourth power plus two. Substituting in 𝑧 is equal to root
𝑥 into our integrand, we get three times root 𝑥 squared divided by root 𝑥 to the
fourth power plus two. And remember, we need to multiply
this by the derivative of root 𝑥 with respect to 𝑥.
However, we can also just evaluate
this derivative by using the power rule for differentiation. We get the derivative of root 𝑥
with respect to 𝑥 is equal to one over two root 𝑥. So we now have that ℎ prime of 𝑥
is equal to three root 𝑥 squared divided by root 𝑥 to the fourth power plus two
multiplied by one over two root 𝑥. And now we can start simplifying
this expression.
First, we see our numerator and our
denominator share a factor of root 𝑥. Next, we see root 𝑥 all raised to
the fourth power is equal to 𝑥 squared. So by using these simplifications
and rearranging, we get three root 𝑥 divided by two times 𝑥 squared plus two. And this is our final answer. Therefore, by using the fundamental
theorem of calculus on the function ℎ of 𝑥 is equal to the integral from four to
root 𝑥 of three 𝑧 squared divided by 𝑧 to the fourth power plus two with respect
to 𝑧, we were able to show that ℎ prime of 𝑥 is equal to three root 𝑥 divided by
two times 𝑥 squared plus two.
