<s>[INST] Find $f'(1)$ given $f(x)$ and $f(1)$ Question:

Let $$f(x)=\int\frac{x-1}{x+1}\frac{dx}{\sqrt{x^3+x^2+x}}$$ such that $f(1)=\frac{2\pi}{3}$
Then $f'(1)$ is equal to
A) $0$
B) $\cfrac \pi3$
C) $\cfrac\pi4$
D) $2$

Attempt:

\begin{align}f(x)&=\int\frac{x-1}{x+1}\frac{1}{\sqrt{x^3+x^2+x}}dx\\
f(1)&=\int0dx\\
f(1)&=0+c\\
&= \frac{2\pi}{3}\\
\implies c&= \frac{2\pi}{3}
\end{align}

I don't know what to do beyond this step. This is an indefinite integral so we cannot differentiate it. How else we bring $f'(x)$ into picture? Any hints?
 [/INST] $f'(x)$ is simply the term under integration.$$f'(x)=\frac{x-1}{x+1}\frac{1}{\sqrt{x + x^2 + x^3}}$$
Take an example.
$$\text{Let} ~~~g(x) = \int xdx ; ~~\text{find g(1), given g(0)=0.}$$.
As you have done , it should be $$g(1) = \int 1dx= x+c$$ using $ Since g(0)=0  \implies c=0$
So, $g(1)$ should $x$, isn't it weird?
Yes, it is, because it is wrong.
By integrating , we will get $$g(x)=\frac{x^2}{2} \implies g(1)=\frac12$$
Therefore, $$f'(1)=\frac{1-1}{1+1}\frac{1}{\sqrt{1 + 1^2 + 1^3}}=0$$
Remember, if :
$$f(x)=\int g(x)dx \implies f'(x)=g(x) ~; ~~$$
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