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Calculus
Derivatives
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Find the derivative of \(\displaystyle y = \frac{- 5 x - 4}{\ln{\left(x \right)}}\).

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

Download \(\LaTeX\) 

\begin{question}Find the derivative of $y = \frac{- 5 x - 4}{\ln{\left(x \right)}}$. 
    \soln{9cm}{Using the quotient rule with $f = - 5 x - 4$, $f' = -5$, $g = \ln{\left(x \right)}$, and $g'= \frac{1}{x}$ gives:\newline 
 \begin{equation*} \frac{ (\ln{\left(x \right)})(-5) - (- 5 x - 4)(\frac{1}{x})}{(\ln{\left(x \right)})^2} = - \frac{5 x \ln{\left(x \right)} - 5 x - 4}{x \ln{\left(x \right)}^{2}} \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 = \frac{- 5 x - 4}{\ln{\left(x \right)}} " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%5Cfrac%7B-%205%20x%20-%204%7D%7B%5Cln%7B%5Cleft%28x%20%5Cright%29%7D%7D%20" alt="LaTeX:  \displaystyle y = \frac{- 5 x - 4}{\ln{\left(x \right)}} " data-equation-content=" \displaystyle y = \frac{- 5 x - 4}{\ln{\left(x \right)}} " /> . </p> </p>
HTML for Canvas
<p> <p>Using the quotient rule with  <img class="equation_image" title=" \displaystyle f = - 5 x - 4 " src="/equation_images/%20%5Cdisplaystyle%20f%20%3D%20-%205%20x%20-%204%20" alt="LaTeX:  \displaystyle f = - 5 x - 4 " data-equation-content=" \displaystyle f = - 5 x - 4 " /> ,  <img class="equation_image" title=" \displaystyle f' = -5 " src="/equation_images/%20%5Cdisplaystyle%20f%27%20%3D%20-5%20" alt="LaTeX:  \displaystyle f' = -5 " data-equation-content=" \displaystyle f' = -5 " /> ,  <img class="equation_image" title=" \displaystyle g = \ln{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cln%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle g = \ln{\left(x \right)} " data-equation-content=" \displaystyle g = \ln{\left(x \right)} " /> , and  <img class="equation_image" title=" \displaystyle g'= \frac{1}{x} " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%20%5Cfrac%7B1%7D%7Bx%7D%20" alt="LaTeX:  \displaystyle g'= \frac{1}{x} " data-equation-content=" \displaystyle g'= \frac{1}{x} " />  gives:<br> 
  <img class="equation_image" title="  \frac{ (\ln{\left(x \right)})(-5) - (- 5 x - 4)(\frac{1}{x})}{(\ln{\left(x \right)})^2} = - \frac{5 x \ln{\left(x \right)} - 5 x - 4}{x \ln{\left(x \right)}^{2}}  " src="/equation_images/%20%20%5Cfrac%7B%20%28%5Cln%7B%5Cleft%28x%20%5Cright%29%7D%29%28-5%29%20-%20%28-%205%20x%20-%204%29%28%5Cfrac%7B1%7D%7Bx%7D%29%7D%7B%28%5Cln%7B%5Cleft%28x%20%5Cright%29%7D%29%5E2%7D%20%3D%20-%20%5Cfrac%7B5%20x%20%5Cln%7B%5Cleft%28x%20%5Cright%29%7D%20-%205%20x%20-%204%7D%7Bx%20%5Cln%7B%5Cleft%28x%20%5Cright%29%7D%5E%7B2%7D%7D%20%20" alt="LaTeX:   \frac{ (\ln{\left(x \right)})(-5) - (- 5 x - 4)(\frac{1}{x})}{(\ln{\left(x \right)})^2} = - \frac{5 x \ln{\left(x \right)} - 5 x - 4}{x \ln{\left(x \right)}^{2}}  " data-equation-content="  \frac{ (\ln{\left(x \right)})(-5) - (- 5 x - 4)(\frac{1}{x})}{(\ln{\left(x \right)})^2} = - \frac{5 x \ln{\left(x \right)} - 5 x - 4}{x \ln{\left(x \right)}^{2}}  " /> </p> </p>