Answer
$f'(x) = g'(x)~~~$ for all $x$ in their domains.
We can not conclude from Corollary 7 that $f- g$ is constant.
Work Step by Step
The domain of $f$ is $(-\infty,0) \cup (0,\infty)$
The domain of $g$ is $(-\infty,0) \cup (0,\infty)$
$f'(x) = -\frac{1}{x^2}$ and $g'(x) = -\frac{1}{x^2}$
Thus $~~~f'(x) = g'(x)~~~$ for all $x$ in their domains.
According to Corollary 7, in the interval $(-\infty,0)$:
$f- g = c_1~~$ for some constant $c_1$
Also, according to Corollary 7, in the interval $(0,\infty)$:
$f- g = c_2~~$ for some constant $c_2$
However, it is possible that $c_1 \neq c_2$
Therefore, we can not conclude from Corollary 7 that $f- g$ is constant.
