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Calculus
Derivatives
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Find the derivative of \(\displaystyle y = \left(- 2 x^{2} + 5 x + 6\right) \ln{\left(x \right)}\).

Using the product rule with \(\displaystyle f = \ln{\left(x \right)}\), \(\displaystyle f' = \frac{1}{x}\), \(\displaystyle g = - 2 x^{2} + 5 x + 6\), and \(\displaystyle g'= 5 - 4 x\) gives:
\begin{equation*} y' = (- 2 x^{2} + 5 x + 6)(\frac{1}{x}) + (\ln{\left(x \right)})(5 - 4 x) = \left(5 - 4 x\right) \ln{\left(x \right)} + \frac{- 2 x^{2} + 5 x + 6}{x} \end{equation*}

Download \(\LaTeX\) 

\begin{question}Find the derivative of $y = \left(- 2 x^{2} + 5 x + 6\right) \ln{\left(x \right)}$. 
    \soln{9cm}{Using the product rule with $f = \ln{\left(x \right)}$, $f' = \frac{1}{x}$, $g = - 2 x^{2} + 5 x + 6$, and $g'= 5 - 4 x$ gives:\newline 
 \begin{equation*} y' =  (- 2 x^{2} + 5 x + 6)(\frac{1}{x}) + (\ln{\left(x \right)})(5 - 4 x) = \left(5 - 4 x\right) \ln{\left(x \right)} + \frac{- 2 x^{2} + 5 x + 6}{x} \end{equation*}}

\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(- 2 x^{2} + 5 x + 6\right) \ln{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%5Cleft%28-%202%20x%5E%7B2%7D%20%2B%205%20x%20%2B%206%5Cright%29%20%5Cln%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle y = \left(- 2 x^{2} + 5 x + 6\right) \ln{\left(x \right)} " data-equation-content=" \displaystyle y = \left(- 2 x^{2} + 5 x + 6\right) \ln{\left(x \right)} " /> . </p> </p>
HTML for Canvas
<p> <p>Using the product rule with  <img class="equation_image" title=" \displaystyle f = \ln{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20f%20%3D%20%5Cln%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle f = \ln{\left(x \right)} " data-equation-content=" \displaystyle f = \ln{\left(x \right)} " /> ,  <img class="equation_image" title=" \displaystyle f' = \frac{1}{x} " src="/equation_images/%20%5Cdisplaystyle%20f%27%20%3D%20%5Cfrac%7B1%7D%7Bx%7D%20" alt="LaTeX:  \displaystyle f' = \frac{1}{x} " data-equation-content=" \displaystyle f' = \frac{1}{x} " /> ,  <img class="equation_image" title=" \displaystyle g = - 2 x^{2} + 5 x + 6 " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20-%202%20x%5E%7B2%7D%20%2B%205%20x%20%2B%206%20" alt="LaTeX:  \displaystyle g = - 2 x^{2} + 5 x + 6 " data-equation-content=" \displaystyle g = - 2 x^{2} + 5 x + 6 " /> , and  <img class="equation_image" title=" \displaystyle g'= 5 - 4 x " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%205%20-%204%20x%20" alt="LaTeX:  \displaystyle g'= 5 - 4 x " data-equation-content=" \displaystyle g'= 5 - 4 x " />  gives:<br> 
  <img class="equation_image" title="  y' =  (- 2 x^{2} + 5 x + 6)(\frac{1}{x}) + (\ln{\left(x \right)})(5 - 4 x) = \left(5 - 4 x\right) \ln{\left(x \right)} + \frac{- 2 x^{2} + 5 x + 6}{x}  " src="/equation_images/%20%20y%27%20%3D%20%20%28-%202%20x%5E%7B2%7D%20%2B%205%20x%20%2B%206%29%28%5Cfrac%7B1%7D%7Bx%7D%29%20%2B%20%28%5Cln%7B%5Cleft%28x%20%5Cright%29%7D%29%285%20-%204%20x%29%20%3D%20%5Cleft%285%20-%204%20x%5Cright%29%20%5Cln%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cfrac%7B-%202%20x%5E%7B2%7D%20%2B%205%20x%20%2B%206%7D%7Bx%7D%20%20" alt="LaTeX:   y' =  (- 2 x^{2} + 5 x + 6)(\frac{1}{x}) + (\ln{\left(x \right)})(5 - 4 x) = \left(5 - 4 x\right) \ln{\left(x \right)} + \frac{- 2 x^{2} + 5 x + 6}{x}  " data-equation-content="  y' =  (- 2 x^{2} + 5 x + 6)(\frac{1}{x}) + (\ln{\left(x \right)})(5 - 4 x) = \left(5 - 4 x\right) \ln{\left(x \right)} + \frac{- 2 x^{2} + 5 x + 6}{x}  " /> </p> </p>