Answer
$$
f^{\prime }(x)=1+3\sqrt {x}
$$
The required function is
$$
f(x) =x+2 x^{3 / 2}+5
$$
Work Step by Step
$$
f^{\prime }(x)=1+3\sqrt {x}
$$
The general anti-derivative of $
f^{\prime }(x)=1+3\sqrt {x}
$ is
$$
\begin{split}
f(x)&=x+3\left(\frac{2}{3} x^{3 / 2}\right)+C \\
& =x+2 x^{3 / 2}+C
\end{split}
$$
To determine C we use the fact that $f(4)=25$:
$$
f(x)=(4)+2(4)^{3 / 2}+C=25
$$
$ \Rightarrow $
$$
20+C=25 \quad \Rightarrow \quad C=5,
$$
so the required function is
$$
f(x) =x+2 x^{3 / 2}+5
$$
