TITLE: Trying to find a function whose derivative is $\cfrac{4x^3}{x^4+1}$ any ideas?
QUESTION [6 upvotes]: The closet I can get is
$\arctan(x^4) $
the derivative of $\arctan(x^4)$ is $\dfrac{4x^3}{x^8 + 1} ...$
Any tips?

REPLY [10 votes]: Caveat: The following explanation is for those who still have no knowledge about Integration/Anti-Derivative. This example emphasizes the fact of approaching an Integration with just the knowledge of differentiation
What you are trying to determine is called anti-derivative or Integration. In view of that consider the detailed explanation of how you should look over the problem.
What we know
$$\frac{df(x)}{dx}=\frac{4x^3}{1+x^4}\tag1$$
We need to determine what $f(x)$ is?
Lets rewrite $(1)$ as
$$df(x)=\frac{4x^3}{1+x^4}dx$$
Lets substitute $y = 1+x^4, dy = 4x^3dx$, then we have
$$df(x)=\frac{dy}{y}\tag2$$
Now from your knowledge of derivative, you know that 
$$\frac{d}{dy}(\log y +C) = \frac{1}{y}$$
$$d(\log y +C) = \frac{dy}{y}\tag3$$
substituting $(3)\text{ on }(2)$
$$df(x)=d(\log(y) +C)$$
Substitute back $y = 1+x^4$
$$df(x)=d(\log(1+x^4)+C)$$
So $f(x)=\log(1+x^4)+C$