Question

Please use the derivative to show why f(x)=integral 1/sqrt(4+e^t)dt from 0 to lnx, with x>0, has an inverse function

Answer 1

did you find the derivative?

Answer 2

maybe i should ask "do you know how to find the derivative?"

Answer 3

no, sorry, I haven't found the derivative.

Answer 4

I actually don't know how to find the derivative.

Answer 5

the derivative of the integral is the integrand
that, and the chain rule

Answer 6

\[\frac{d}{dx}\int_a^x f(t)dt=f(x)\]

Answer 7

\[\frac{d}{dx}\int_a^{g(x)}f(t)dt=f(g(x))g'(x)\]

Answer 8

you have
\[\int_0^{\ln(x)}\frac{1}{\sqrt{4+e^t}}dt\] if i am reading it correctly

Answer 9

replace \(t\) by \(\ln(x)\) in \(\frac{1}{\sqrt{4+e^t}}\) and multiply by \(\frac{1}{x}\) to get the derivative

Answer 10

that makes your derivative
\[\large \frac{1}{\sqrt{4+e^{\ln(x)}}}\times \frac{1}{x}\] or more simply
\[\frac{1}{x\sqrt{4+x}}\]

Answer 11

Thank you so much! that makes a lot of sense. I actually understand how to do it now!