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Calculus
Derivatives
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Find the derivative \(\displaystyle f(x) = \frac{- 5 x^{\frac{2}{3}} - 4 x^{\frac{1}{3}} + 2 x}{x^{\frac{1}{3}}}\)

Using termwise division gives \(\displaystyle f(x) = 2 x^{\frac{2}{3}} - 5 x^{\frac{1}{3}} - 4\). Now the power rule for derivatives can be used instead of the quotient rule. This gives \begin{equation*}f'(x) = \frac{4}{3 \sqrt[3]{x}} - \frac{5}{3 x^{\frac{2}{3}}} \end{equation*}

Download \(\LaTeX\) 

\begin{question}Find the derivative $f(x) = \frac{- 5 x^{\frac{2}{3}} - 4 x^{\frac{1}{3}} + 2 x}{x^{\frac{1}{3}}}$
    \soln{9cm}{Using termwise division gives $f(x) = 2 x^{\frac{2}{3}} - 5 x^{\frac{1}{3}} - 4$.  Now the power rule for derivatives can be used instead of the quotient rule. This gives
\begin{equation*}f'(x) = \frac{4}{3 \sqrt[3]{x}} - \frac{5}{3 x^{\frac{2}{3}}} \end{equation*}}

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Find the derivative  <img class="equation_image" title=" \displaystyle f(x) = \frac{- 5 x^{\frac{2}{3}} - 4 x^{\frac{1}{3}} + 2 x}{x^{\frac{1}{3}}} " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20%3D%20%5Cfrac%7B-%205%20x%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%20-%204%20x%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%20%2B%202%20x%7D%7Bx%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%7D%20" alt="LaTeX:  \displaystyle f(x) = \frac{- 5 x^{\frac{2}{3}} - 4 x^{\frac{1}{3}} + 2 x}{x^{\frac{1}{3}}} " data-equation-content=" \displaystyle f(x) = \frac{- 5 x^{\frac{2}{3}} - 4 x^{\frac{1}{3}} + 2 x}{x^{\frac{1}{3}}} " /> </p> </p>
HTML for Canvas
<p> <p>Using termwise division gives  <img class="equation_image" title=" \displaystyle f(x) = 2 x^{\frac{2}{3}} - 5 x^{\frac{1}{3}} - 4 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20%3D%202%20x%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%20-%205%20x%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%20-%204%20" alt="LaTeX:  \displaystyle f(x) = 2 x^{\frac{2}{3}} - 5 x^{\frac{1}{3}} - 4 " data-equation-content=" \displaystyle f(x) = 2 x^{\frac{2}{3}} - 5 x^{\frac{1}{3}} - 4 " /> .  Now the power rule for derivatives can be used instead of the quotient rule. This gives
 <img class="equation_image" title=" f'(x) = \frac{4}{3 \sqrt[3]{x}} - \frac{5}{3 x^{\frac{2}{3}}}  " src="/equation_images/%20f%27%28x%29%20%3D%20%5Cfrac%7B4%7D%7B3%20%5Csqrt%5B3%5D%7Bx%7D%7D%20-%20%5Cfrac%7B5%7D%7B3%20x%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%7D%20%20" alt="LaTeX:  f'(x) = \frac{4}{3 \sqrt[3]{x}} - \frac{5}{3 x^{\frac{2}{3}}}  " data-equation-content=" f'(x) = \frac{4}{3 \sqrt[3]{x}} - \frac{5}{3 x^{\frac{2}{3}}}  " /> </p> </p>