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.