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Find the derivative \(\displaystyle f(x) = \frac{- 3 x^{\frac{7}{3}} - 2 x^{\frac{4}{3}} + 2 x^{\frac{1}{3}}}{x^{\frac{1}{3}}}\)

Using termwise division gives \(\displaystyle f(x) = - 3 x^{2} - 2 x + 2\). Now the power rule for derivatives can be used instead of the quotient rule. This gives \begin{equation*}f'(x) = - 6 x - 2 \end{equation*}

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\begin{question}Find the derivative $f(x) = \frac{- 3 x^{\frac{7}{3}} - 2 x^{\frac{4}{3}} + 2 x^{\frac{1}{3}}}{x^{\frac{1}{3}}}$
    \soln{9cm}{Using termwise division gives $f(x) = - 3 x^{2} - 2 x + 2$.  Now the power rule for derivatives can be used instead of the quotient rule. This gives
\begin{equation*}f'(x) = - 6 x - 2 \end{equation*}}

\end{question}

Download Question and Solution Environment\(\LaTeX\)

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Find the derivative <img class="equation_image" title=" \displaystyle f(x) = \frac{- 3 x^{\frac{7}{3}} - 2 x^{\frac{4}{3}} + 2 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20%3D%20%5Cfrac%7B-%203%20x%5E%7B%5Cfrac%7B7%7D%7B3%7D%7D%20-%202%20x%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D%20%2B%202%20x%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%7D%7Bx%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%7D%20" alt="LaTeX: \displaystyle f(x) = \frac{- 3 x^{\frac{7}{3}} - 2 x^{\frac{4}{3}} + 2 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} " data-equation-content=" \displaystyle f(x) = \frac{- 3 x^{\frac{7}{3}} - 2 x^{\frac{4}{3}} + 2 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} " />

Using termwise division gives <img class="equation_image" title=" \displaystyle f(x) = - 3 x^{2} - 2 x + 2 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20%3D%20-%203%20x%5E%7B2%7D%20-%202%20x%20%2B%202%20" alt="LaTeX: \displaystyle f(x) = - 3 x^{2} - 2 x + 2 " data-equation-content=" \displaystyle f(x) = - 3 x^{2} - 2 x + 2 " /> . Now the power rule for derivatives can be used instead of the quotient rule. This gives <img class="equation_image" title=" f'(x) = - 6 x - 2 " src="/equation_images/%20f%27%28x%29%20%3D%20-%206%20x%20-%202%20%20" alt="LaTeX: f'(x) = - 6 x - 2 " data-equation-content=" f'(x) = - 6 x - 2 " />