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Calculus
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Find the derivative of \(\displaystyle y=\left(x^{3} - 7 x^{2} - 11 x + 5\right)^{2}\).

The outer function is \(\displaystyle f(u) = u^{2}\) and the inner function \(\displaystyle u = x^{3} - 7 x^{2} - 11 x + 5\). The chain rule gives \(\displaystyle y'= \frac{df}{du}\frac{du}{dx}\). \(\displaystyle y' = 2 u(3 x^{2} - 14 x - 11) = \left(6 x^{2} - 28 x - 22\right) \left(x^{3} - 7 x^{2} - 11 x + 5\right)\). Simplifying further gives \(\displaystyle y'= 6 x^{5} - 70 x^{4} + 108 x^{3} + 492 x^{2} + 102 x - 110\)

Download \(\LaTeX\) 

\begin{question}Find the derivative of $y=\left(x^{3} - 7 x^{2} - 11 x + 5\right)^{2}$. 
    \soln{9cm}{The outer function is $f(u) = u^{2}$ and the inner function $u = x^{3} - 7 x^{2} - 11 x + 5$. The chain rule gives $y'= \frac{df}{du}\frac{du}{dx}$. $y' = 2 u(3 x^{2} - 14 x - 11) = \left(6 x^{2} - 28 x - 22\right) \left(x^{3} - 7 x^{2} - 11 x + 5\right)$. Simplifying further gives $y'= 6 x^{5} - 70 x^{4} + 108 x^{3} + 492 x^{2} + 102 x - 110$}

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Find the derivative of  <img class="equation_image" title=" \displaystyle y=\left(x^{3} - 7 x^{2} - 11 x + 5\right)^{2} " src="/equation_images/%20%5Cdisplaystyle%20y%3D%5Cleft%28x%5E%7B3%7D%20-%207%20x%5E%7B2%7D%20-%2011%20x%20%2B%205%5Cright%29%5E%7B2%7D%20" alt="LaTeX:  \displaystyle y=\left(x^{3} - 7 x^{2} - 11 x + 5\right)^{2} " data-equation-content=" \displaystyle y=\left(x^{3} - 7 x^{2} - 11 x + 5\right)^{2} " /> . </p> </p>
HTML for Canvas
<p> <p>The outer function is  <img class="equation_image" title=" \displaystyle f(u) = u^{2} " src="/equation_images/%20%5Cdisplaystyle%20f%28u%29%20%3D%20u%5E%7B2%7D%20" alt="LaTeX:  \displaystyle f(u) = u^{2} " data-equation-content=" \displaystyle f(u) = u^{2} " />  and the inner function  <img class="equation_image" title=" \displaystyle u = x^{3} - 7 x^{2} - 11 x + 5 " src="/equation_images/%20%5Cdisplaystyle%20u%20%3D%20x%5E%7B3%7D%20-%207%20x%5E%7B2%7D%20-%2011%20x%20%2B%205%20" alt="LaTeX:  \displaystyle u = x^{3} - 7 x^{2} - 11 x + 5 " data-equation-content=" \displaystyle u = x^{3} - 7 x^{2} - 11 x + 5 " /> . 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' = 2 u(3 x^{2} - 14 x - 11) = \left(6 x^{2} - 28 x - 22\right) \left(x^{3} - 7 x^{2} - 11 x + 5\right) " src="/equation_images/%20%5Cdisplaystyle%20y%27%20%3D%202%20u%283%20x%5E%7B2%7D%20-%2014%20x%20-%2011%29%20%3D%20%5Cleft%286%20x%5E%7B2%7D%20-%2028%20x%20-%2022%5Cright%29%20%5Cleft%28x%5E%7B3%7D%20-%207%20x%5E%7B2%7D%20-%2011%20x%20%2B%205%5Cright%29%20" alt="LaTeX:  \displaystyle y' = 2 u(3 x^{2} - 14 x - 11) = \left(6 x^{2} - 28 x - 22\right) \left(x^{3} - 7 x^{2} - 11 x + 5\right) " data-equation-content=" \displaystyle y' = 2 u(3 x^{2} - 14 x - 11) = \left(6 x^{2} - 28 x - 22\right) \left(x^{3} - 7 x^{2} - 11 x + 5\right) " /> . Simplifying further gives  <img class="equation_image" title=" \displaystyle y'= 6 x^{5} - 70 x^{4} + 108 x^{3} + 492 x^{2} + 102 x - 110 " src="/equation_images/%20%5Cdisplaystyle%20y%27%3D%206%20x%5E%7B5%7D%20-%2070%20x%5E%7B4%7D%20%2B%20108%20x%5E%7B3%7D%20%2B%20492%20x%5E%7B2%7D%20%2B%20102%20x%20-%20110%20" alt="LaTeX:  \displaystyle y'= 6 x^{5} - 70 x^{4} + 108 x^{3} + 492 x^{2} + 102 x - 110 " data-equation-content=" \displaystyle y'= 6 x^{5} - 70 x^{4} + 108 x^{3} + 492 x^{2} + 102 x - 110 " /> </p> </p>