Question

Find the derivative of the function....

Answer 1

\[F(x)=\int\limits_{1}^{\sqrt{x}}\frac{ t ^{2} }{ t ^{2}+1 }\]

Answer 2

the derivative of the integral is the integrand, plus chain rule
replace \(t\) by \(\sqrt{x}\) and the via the chain rule multiply by \(\frac{1}{2\sqrt{x}}\)

Answer 3

clear?

Answer 4

not completely that is the part I'm having the issue with. what I want to do is this t^2(t^2+1)^-1

Answer 5

whoa back up

Answer 6

\[f(t)=\frac{ t ^{2} }{ t ^{2}+1 }\] replace \(t\) by \(\sqrt{x}\)

Answer 7

yeah my mind want the denominator on top...

Answer 8

\[\frac{d}{dx}\int _a^xf(t)dt=f(x)\] for example
\[\frac{d}{dx}\int _a^x\sin(t)dt=\sin(x)\]and
\[\frac{d}{dx}\int _a^x\frac{t}{t-1}dt=\frac{x}{x-1}\]

Answer 9

so replace t with x

Answer 10

the only difference in my example and yours is that the upper limit of integration is not \(x\) it is \(\sqrt{x}\) making this a composite function
replace \(t\) by \(\sqrt{x}\) is all

Answer 11

then by the chain rule, multiply by the derivative of \(\sqrt{x}\) which is \(\frac{1}{2\sqrt{x}}\)

Answer 12

so it if was just
\[F(x)=\int_1^x\frac{t^2}{t^2+1}dt\] then the answer would be
\[F'(x)=\frac{x^2}{x^2+1}\] but this is a composition, it is like
\[F(\sqrt{x})\] so the derivative is
\[\frac{\sqrt{x}^2}{\sqrt{x}^2+1}\times \frac{1}{2\sqrt{x}}\] which of course cleans up a lot via algebra

Answer 13

i have \[\frac{ x }{ 2x \sqrt{x} +2\sqrt{x}}\] from that

Answer 14

i guess

Answer 15

do I need to break it down more I'm not sure.

Answer 16

i would have left it as
\[\frac{x}{x+1}\times \frac{1}{2\sqrt{x}}\] but you are right

Answer 17

your answer looks good
i guess you could do even more algebra, but who cares?

Answer 18

actually nothing useful
leave it as that
hope the idea is clear

Answer 19

\[F(x)=\int_1^{\sin(x)}e^tdt\]
\[F'(x)=e^{\sin(x)}\times \cos(x)\] is another example

Answer 20

Yes, I definitely get it now.

Answer 21

k good
and also easy right? when you know what you are doing
otherwise it is very confusing, like you have to evaluate the integral or something

Answer 22

exactly