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Calculus
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Find the derivative of \(\displaystyle y=\sqrt[4]{\tan^{2}{\left(x \right)}}\).

The outer function is \(\displaystyle f(u) = \sqrt[4]{u^{2}}\) and the inner function \(\displaystyle u = \tan{\left(x \right)}\). The chain rule gives \(\displaystyle y'= \frac{df}{du}\frac{du}{dx}\). \(\displaystyle y' = \frac{\sqrt[4]{u^{2}}}{2 u}(\tan^{2}{\left(x \right)} + 1) = \frac{\left(2 \tan^{2}{\left(x \right)} + 2\right) \sqrt[4]{\tan^{2}{\left(x \right)}}}{4 \tan{\left(x \right)}}\). Simplifying further gives \(\displaystyle y'= \frac{\sqrt[4]{\tan^{2}{\left(x \right)}}}{\sin{\left(2 x \right)}}\)

Download \(\LaTeX\) 

\begin{question}Find the derivative of $y=\sqrt[4]{\tan^{2}{\left(x \right)}}$. 
    \soln{9cm}{The outer function is $f(u) = \sqrt[4]{u^{2}}$ and the inner function $u = \tan{\left(x \right)}$. The chain rule gives $y'= \frac{df}{du}\frac{du}{dx}$. $y' = \frac{\sqrt[4]{u^{2}}}{2 u}(\tan^{2}{\left(x \right)} + 1) = \frac{\left(2 \tan^{2}{\left(x \right)} + 2\right) \sqrt[4]{\tan^{2}{\left(x \right)}}}{4 \tan{\left(x \right)}}$. Simplifying further gives $y'= \frac{\sqrt[4]{\tan^{2}{\left(x \right)}}}{\sin{\left(2 x \right)}}$}

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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\begin{document}\begin{question}(10pts) The question goes here!
    \soln{9cm}{The solution goes here.}

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HTML for Canvas 
<p> <p>Find the derivative of  <img class="equation_image" title=" \displaystyle y=\sqrt[4]{\tan^{2}{\left(x \right)}} " src="/equation_images/%20%5Cdisplaystyle%20y%3D%5Csqrt%5B4%5D%7B%5Ctan%5E%7B2%7D%7B%5Cleft%28x%20%5Cright%29%7D%7D%20" alt="LaTeX:  \displaystyle y=\sqrt[4]{\tan^{2}{\left(x \right)}} " data-equation-content=" \displaystyle y=\sqrt[4]{\tan^{2}{\left(x \right)}} " /> . </p> </p>
HTML for Canvas
<p> <p>The outer function is  <img class="equation_image" title=" \displaystyle f(u) = \sqrt[4]{u^{2}} " src="/equation_images/%20%5Cdisplaystyle%20f%28u%29%20%3D%20%5Csqrt%5B4%5D%7Bu%5E%7B2%7D%7D%20" alt="LaTeX:  \displaystyle f(u) = \sqrt[4]{u^{2}} " data-equation-content=" \displaystyle f(u) = \sqrt[4]{u^{2}} " />  and the inner function  <img class="equation_image" title=" \displaystyle u = \tan{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20u%20%3D%20%5Ctan%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle u = \tan{\left(x \right)} " data-equation-content=" \displaystyle u = \tan{\left(x \right)} " /> . The chain rule gives  <img class="equation_image" title=" \displaystyle y'= \frac{df}{du}\frac{du}{dx} " src="/equation_images/%20%5Cdisplaystyle%20y%27%3D%20%5Cfrac%7Bdf%7D%7Bdu%7D%5Cfrac%7Bdu%7D%7Bdx%7D%20" alt="LaTeX:  \displaystyle y'= \frac{df}{du}\frac{du}{dx} " data-equation-content=" \displaystyle y'= \frac{df}{du}\frac{du}{dx} " /> .  <img class="equation_image" title=" \displaystyle y' = \frac{\sqrt[4]{u^{2}}}{2 u}(\tan^{2}{\left(x \right)} + 1) = \frac{\left(2 \tan^{2}{\left(x \right)} + 2\right) \sqrt[4]{\tan^{2}{\left(x \right)}}}{4 \tan{\left(x \right)}} " src="/equation_images/%20%5Cdisplaystyle%20y%27%20%3D%20%5Cfrac%7B%5Csqrt%5B4%5D%7Bu%5E%7B2%7D%7D%7D%7B2%20u%7D%28%5Ctan%5E%7B2%7D%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%201%29%20%3D%20%5Cfrac%7B%5Cleft%282%20%5Ctan%5E%7B2%7D%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%202%5Cright%29%20%5Csqrt%5B4%5D%7B%5Ctan%5E%7B2%7D%7B%5Cleft%28x%20%5Cright%29%7D%7D%7D%7B4%20%5Ctan%7B%5Cleft%28x%20%5Cright%29%7D%7D%20" alt="LaTeX:  \displaystyle y' = \frac{\sqrt[4]{u^{2}}}{2 u}(\tan^{2}{\left(x \right)} + 1) = \frac{\left(2 \tan^{2}{\left(x \right)} + 2\right) \sqrt[4]{\tan^{2}{\left(x \right)}}}{4 \tan{\left(x \right)}} " data-equation-content=" \displaystyle y' = \frac{\sqrt[4]{u^{2}}}{2 u}(\tan^{2}{\left(x \right)} + 1) = \frac{\left(2 \tan^{2}{\left(x \right)} + 2\right) \sqrt[4]{\tan^{2}{\left(x \right)}}}{4 \tan{\left(x \right)}} " /> . Simplifying further gives  <img class="equation_image" title=" \displaystyle y'= \frac{\sqrt[4]{\tan^{2}{\left(x \right)}}}{\sin{\left(2 x \right)}} " src="/equation_images/%20%5Cdisplaystyle%20y%27%3D%20%5Cfrac%7B%5Csqrt%5B4%5D%7B%5Ctan%5E%7B2%7D%7B%5Cleft%28x%20%5Cright%29%7D%7D%7D%7B%5Csin%7B%5Cleft%282%20x%20%5Cright%29%7D%7D%20" alt="LaTeX:  \displaystyle y'= \frac{\sqrt[4]{\tan^{2}{\left(x \right)}}}{\sin{\left(2 x \right)}} " data-equation-content=" \displaystyle y'= \frac{\sqrt[4]{\tan^{2}{\left(x \right)}}}{\sin{\left(2 x \right)}} " /> </p> </p>