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%  Differentiation Rules: General Power Rule                       %
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\problemgrouptitle{Differentiation Rules: General Power Rule}

\begin{problemblock}[2]{Differentiate the following functions requiring use of the General Power Rule. (Any value that is not the function variable should be considered a constant.)}

\item $\displaystyle f(x)=\left(x^2+3\right)^9$
\answer{$\displaystyle f'(x)=18x\left(x^2+3\right)^8$}

\item $\displaystyle g(x)=\frac{1}{x+\sqrt{x}}$
\answer{$\displaystyle \frac{dg}{dx}=-\frac{\frac{1}{2\sqrt{x}}+1}{\left(x+\sqrt{x}\right)^2}=-\frac{2\sqrt{x}+1}{2\sqrt{x}\left(x^2+x\right)+4x^2}$}

\item $\displaystyle f(t)=\frac{7}{\sqrt{2t^2+3t+4}}$
\answer{$\displaystyle f'(t)=-\frac{7(4t+3)}{2\left(2t^2+3t+4\right)^{\frac{3}{2}}}$}

\item $\displaystyle y=\left(\frac{4x+3}{x^2+x}\right)^{-\frac{1}{7}}$
\answer{$\displaystyle y'=\frac{1}{7}\left(\frac{4x+3}{x^2+x}\right)^{-\frac{8}{7}}\frac{4x^2+6x+3}{\left(x^2+x\right)^2}$}

\item $\displaystyle h(x)=\sqrt[3]{5x^n+4c}$
\answer{$\displaystyle h'(x)=\frac{5nx^{n-1}}{3\left(5x^n+4c\right)^{\frac{2}{3}}}$}

\item $\displaystyle f(x)=\left[\left(2x+\sqrt{x}\,\right)^4+3x\right]^5$
\answer{$\displaystyle f'(x)=5\left[{\left(2\,x+\sqrt{x}\,\right) }^{4}+3\,x\right]^4\left[4\left(2\,x+\sqrt{x}\right)^3\left(2+\frac{1}{2\sqrt{x}}\right)+3\right]$}

\end{problemblock}
