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Studying for a test? Prepare with these 9 lessons on Antiderivatives and the fundamental theorem of calculus.

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# Worked example: Finding derivative with fundamental theorem of calculus

Video transcript

let's say that we have capital F of axes
being equal to the definite integral from pod to acts of coat engine squared
of tea det and will were curious about finding is the derivative of this
business we want to figure out what F prime what F prime of x is equal do well
this is a direct application of the fundamental theorem of calculus this is
going to be the derivative with respect to acts of all of this craziness so let
me just copy and paste that copy and paste so it's just gonna be the
derivative with respect to that and the fundamental theorem of calculus tells us
that this is going to be equal to this is going to be equal to just add this
function this function but it's not going to be a function of TA more it's
going to be a function it's going to be a function of acts so it's going to be
equal to go tangent squared ax they'll often see problems like this if you go
to calculus competitions or things like that and maybe certain exams it's a cold
my god I have have to take the anti derivative of all of this business evaluated at the different boundaries
and then I gotta take the derivative know you just apply the fundamental
theorem of calculus it's actually very straightforward and a very fast thing to
do now let's mix it up a little bit let's say that you had the expression
the deficit in a girl from Popeyes had a from pod to access it from PI two x
squared squared a different color just to make it clear what I'm doing from Popeye to x squared squared Co
tangent coat engine squared of t he T and you want to take the derivative of
this business so you want to take the derivative with respect to axe you won't
take the derivative with respect to acts of this business how would you do it
well the recognition here is is that this is capital at Capital FM X was
defined as this now set of an ex you have a neck square to this is the exact
same thing is taking the derivative with respect to acts of capital and capital F not acts that would be this is ted we
have capital F of X squared where we had an extra before we now have an ex
queried we now have the next grade you can verify if you had capital after of X
squared everybody's we saw the extra would be an expired so would look
exactly like this and so we just have to apply the chain rule so this is going to
be equal to and this is going to be equal to the derivative the derivative
of F with respect to x squared so it's going to be this is a straight out of
the chain rule it's going to be asked prime of X squared x squared times the
derivative of the derivative with respect to ax ax squared this is
derivatives of ass with respect to x squared and then times and river X
squared with respect to access this is just the chain rule right over here so
what's a crime of X where the derivative of F with respect to x squared well if
you evaluate a private X squared instead of just being coat and it's great X is
going to be co tangent squared x squared so this part right over here this
business writer here is going to be co tangent squared x squared x squared so
this is the derivative of all of this business with respect to x squared then
you have to multiply that you have to multiply that times the derivative of X
squared with respect to acts which is just two acts so there you have it the
derivative of this all this craziness is equal to two acts Co tangent squared x
squared