Question

Evaluate \(g^{\prime}(x)\) using the rules for differentiation:
\[
g(x)=-\frac{1}{2} x^{\frac{7}{2}}+\frac{3}{4} \sqrt[4]{x^{3}}-\frac{4}{3} x^{\frac{1}{2}}
\]

Answer 1

\[
\begin{aligned}
& \frac{d}{d x}\left(-\frac{1}{2} x^{\frac{7}{2}}+\frac{3}{4} x^{\frac{3}{4}}-\frac{4}{3} x^{\frac{1}{2}}\right) \\
=&-\left(\frac{7}{2}\right)\left(\frac{1}{2}\right) x^{\left(\frac{7}{2}-1\right)} \\
&+\left(\frac{3}{4}\right)\left(\frac{3}{4}\right) x^{\left(\frac{3}{4}-1\right)} \\
&-\left(\frac{1}{2}\right)\left(\frac{4}{3}\right) x^{\left(\frac{1}{2}-1\right)}
\end{aligned}
\]
We write the final answer with positive exponents:
\[
g^{\prime}(x)=-\frac{7}{4} x^{\frac{5}{2}}+\frac{9}{16 x^{\frac{1}{4}}}-\frac{2}{3 x^{\frac{1}{2}}}
\]