Single Variable Calculus, Chapter 7, 7.6, Section 7.6, Problem 42

Determine $f(x) = \arctan \left(x^2 - x\right)$. Check your answer if it's reasonable by comparing the graphs of $f$ and $f'$.
If $f(x) = \arctan \left(x^2 - x\right)$, then

$\begin{equation} \begin{aligned} f'(x) &= \frac{\frac{d}{dx} \left(x^2 - x\right) }{1 + \left(x^2 - x\right)^2}\\ \\ f'(x) &= \frac{2x-1}{1 + x^4 - 2x^3 + x^2}\\ \\ f'(x) &= \frac{2x-1}{x^4-2x^3 + x^2 +1} \end{aligned} \end{equation}$




We can see from the graph that $f$ is increasing when $f'$ is positive. On the other hand, $f$ is decreasing whenever $f'$ is negative. Thus, we can say that our answer is reasonable.