Implicit differentiation and inverse trigonometric functions practice question 26

Each choice makes a claim about $f'$ and a claim about the values of $f$. Check both, and look at what the domain $x \neq 0$ is made of.

Differentiate $\arctan\!\left(\dfrac{1}{x}\right)$ with the chain rule: the inside $\dfrac{1}{x}$ has derivative $-\dfrac{1}{x^{2}}$. Simplify $f'$.

A function whose derivative is 0 on an interval is constant there. The domain here is two intervals, so evaluate $f$ once on each, for example at $x = 1$ and $x = -1$.