Given,
$\begin{aligned}
f(x) & =\frac{1}{x^3} \int_5^x\left(2 u^2-u f^{\prime}(u) d u\right. \\
\Rightarrow \quad x^3 f(x) & =\int_5^x\left(2 u^2-u f^{\prime}(u)\right) d u
\end{aligned}$
On differentiating w.r.t ' $x$ ',
$\begin{aligned}
x^3 f^{\prime}(x)+3 x^2 f(x) & =2 x^2-x f^{\prime}(x) \\
f^{\prime}(x) & =\frac{2 x^2-3 x^2 f(x)}{x^3+x} \\
f^{\prime}(5) & =\frac{2(5)^2-3(5)^2 f(5)}{5^3+5} \\
f^{\prime}(5) & =\frac{50-0}{130}=\frac{50}{130} \quad[\because f(5)=0] \\
f^{\prime}(5) & =\frac{5}{13}
\end{aligned}$