Answer
$$
f(x)=(x-a)(x-b)(x-c)
$$
By using the Product Rule to differentiate the function $f$ we have :
$$
f^{\prime}(x)=(x-b)(x-c)+(x-a)(x-c)+(x-a)(x-b)
$$
So
$$
\begin{aligned}
\frac{f^{\prime}(x)}{f(x)}&=\frac{(x-b)(x-c)+(x-a)(x-c)+(x-a)(x-b)}{(x-a)(x-b)(x-c)}\\
&=\frac{1}{x-a}+\frac{1}{x-b}+\frac{1}{x-c}.
\end{aligned}
$$
Work Step by Step
$$
f(x)=(x-a)(x-b)(x-c)
$$
By using the Product Rule to differentiate the function $f$ we have :
$$
f^{\prime}(x)=(x-b)(x-c)+(x-a)(x-c)+(x-a)(x-b)
$$
So
$$
\begin{aligned}
\frac{f^{\prime}(x)}{f(x)}&=\frac{(x-b)(x-c)+(x-a)(x-c)+(x-a)(x-b)}{(x-a)(x-b)(x-c)}\\
&=\frac{1}{x-a}+\frac{1}{x-b}+\frac{1}{x-c}.
\end{aligned}
$$
