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

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

Download \(\LaTeX\)

\begin{question}Find the derivative $f(x) = \frac{- 2 x^{\frac{5}{2}} + 4 x^{\frac{3}{2}} - x^{\frac{1}{2}}}{x^{\frac{1}{2}}}$
    \soln{9cm}{Using termwise division gives $f(x) = - 2 x^{2} + 4 x - 1$.  Now the power rule for derivatives can be used instead of the quotient rule. This gives
\begin{equation*}f'(x) = 4 - 4 x \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{- 2 x^{\frac{5}{2}} + 4 x^{\frac{3}{2}} - x^{\frac{1}{2}}}{x^{\frac{1}{2}}} " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20%3D%20%5Cfrac%7B-%202%20x%5E%7B%5Cfrac%7B5%7D%7B2%7D%7D%20%2B%204%20x%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%20-%20x%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D%7Bx%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D%20" alt="LaTeX: \displaystyle f(x) = \frac{- 2 x^{\frac{5}{2}} + 4 x^{\frac{3}{2}} - x^{\frac{1}{2}}}{x^{\frac{1}{2}}} " data-equation-content=" \displaystyle f(x) = \frac{- 2 x^{\frac{5}{2}} + 4 x^{\frac{3}{2}} - x^{\frac{1}{2}}}{x^{\frac{1}{2}}} " />

Using termwise division gives <img class="equation_image" title=" \displaystyle f(x) = - 2 x^{2} + 4 x - 1 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20%3D%20-%202%20x%5E%7B2%7D%20%2B%204%20x%20-%201%20" alt="LaTeX: \displaystyle f(x) = - 2 x^{2} + 4 x - 1 " data-equation-content=" \displaystyle f(x) = - 2 x^{2} + 4 x - 1 " /> . Now the power rule for derivatives can be used instead of the quotient rule. This gives <img class="equation_image" title=" f'(x) = 4 - 4 x " src="/equation_images/%20f%27%28x%29%20%3D%204%20-%204%20x%20%20" alt="LaTeX: f'(x) = 4 - 4 x " data-equation-content=" f'(x) = 4 - 4 x " />