Question

Fundamental Theorem of Calculus:
Find the derivative of h(x)=integral [1,e^x] ln(t) dt

Answer 1

To integrate f(t) from t=1 to t=e^x we get
F(x)=F(e^t)-F(1)

Now to find the derivative of F we simply take the derivative of our above equation.

Answer 2

You must know chain rule.

Answer 3

that one t above is suppose to be an x

Answer 4

F(x)=F(e^x)-F(1)

Answer 5

The derivative of F(x) is of course (x)'*f(x)=1*f(x)=f(x)
Now how do you find the derivative of F(e^x)?

Answer 6

I have no idea what the integral of ln t is.

Answer 7

ok I'm not asking you to do that

Answer 8

I'm asking you to differentiate F(e^x) where F'=f

Answer 9

isn't F supposed to be the antiderivative of the original function?

Answer 10

I let f(t)=ln(t)
I gave the antiderivative of f the name F(t) then I plugged in the upper limit minus plug in the lower limit
giving me F(x)=F(e^x)-F(1)
But we are suppose to be differentiating this so that is what I was asking you to differentiate F(e^x)

Answer 11

\[\int\limits_{a}^{b}f(x) dx=F(x)|_a^b=F(b)-F(a) \text{ where F' =f } \]
I'm using this here in this problem.
\[\frac{d}{dx} (\int\limits_{1}^{e^x}f(t) dt )=\frac{d}{dx}(F(t)|_1^{e^x})=\frac{d}{dx}(F(e^x)-F(1))\]

Answer 12

I'm replacing ln(t) with f(t) for right now
we will come back and use it later

Answer 13

I have to know. Do you understand this so far.

Answer 14

Not really, but to be fair my brain is burned out. I'll have to come back to this, thanks for your time and explanations.

Answer 15

I integrated so I got rid the integral sign

Answer 16

I'm stuck on the formula and it just seems there is a creative approach you're using that's throwing me off. i'll come back to this thanks

Answer 17

It isn't that creative.

Answer 18

Do you have a problem with me calling the antiderivative of f F?

Answer 19

Because that is all I really did so far