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

by Platinum
(106,848 points)
in Mathematics
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Evaluate \(\frac{d}{d x} f(x)\) using the rules for differentiation:
\[
f(x)=-\frac{1}{2} x^{4}-2 x^{3}+\frac{1}{2} \sqrt{x^{5}}
\]

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Best answer
\[
\begin{aligned}
& \frac{d}{d x}\left(-\frac{1}{2} x^{4}-2 x^{3}+\frac{1}{2} x^{\frac{5}{2}}\right) \\
=&-(4)\left(\frac{1}{2}\right) x^{(4-1)} \\
&-(3)(2) x^{(3-1)} \\
&+\left(\frac{5}{2}\right)\left(\frac{1}{2}\right) x^{\left(\frac{5}{2}-1\right)}
\end{aligned}
\]
We write the final answer with positive exponents:
\[
\frac{d}{d x} f(x)=-2 x^{3}-6 x^{2}+\frac{5}{4} x^{\frac{3}{2}}
\]
by Platinum
(106,848 points)

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