Question

Evaluate \(\frac{d}{d x} f(x)\) using the rules for differentiation:
\[
f(x)=-\frac{1}{2} x^{4}-2 x^{3}+\frac{1}{2} \sqrt{x^{5}}
\]

Answer 1

\[
\begin{aligned}
& \frac{d}{d x}\left(-\frac{1}{2} x^{4}-2 x^{3}+\frac{1}{2} x^{\frac{5}{2}}\right) \\
=&-(4)\left(\frac{1}{2}\right) x^{(4-1)} \\
&-(3)(2) x^{(3-1)} \\
&+\left(\frac{5}{2}\right)\left(\frac{1}{2}\right) x^{\left(\frac{5}{2}-1\right)}
\end{aligned}
\]
We write the final answer with positive exponents:
\[
\frac{d}{d x} f(x)=-2 x^{3}-6 x^{2}+\frac{5}{4} x^{\frac{3}{2}}
\]