022 Exam 2 Sample B, Problem 1

Find the derivative of  $y\,=\,\ln {\frac {(x+1)^{4}}{(2x-5)(x+4)}}.$

Foundations:
This problem is best approached through properties of logarithms. Remember that
$\ln(xy)=\ln x+\ln y,$
while
$\ln \left({\frac {x}{y}}\right)=\ln x-\ln y,$
and
$\ln \left(x^{n}\right)=n\ln x,$
You will also need to apply
The Chain Rule: If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} are differentiable functions, then
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (f\circ g)'(x) = f'(g(x))\cdot g'(x).}
Finally, recall that the derivative of natural log is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\ln x\right)'\,=\,\frac{1}{x}.}

 Solution:

Step 1:
We can use the log rules to rewrite our function as
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} y & = & \displaystyle{\ln \frac{(x+1)^4}{(2x - 5)(x + 4)}}\\ \\ & = & 4\ln (x+1)-\ln(2x-5)-\ln (x+4). \end{array}}

Step 2:
We can differentiate term-by-term, applying the chain rule to each term to find
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} y' & = & \displaystyle{4 \cdot \frac{1}{x+1}\cdot(x+1)' - \frac{1}{2x-5}\cdot(2x-5)' - \frac{1}{x+4} \cdot (x+4)'}\\ \\ & = & \displaystyle{\frac{4}{x+1}-\frac{2}{2x-5}-\frac{1}{x+4}}. \end{array}}

Final Answer:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'\,=\,\displaystyle{\frac{4}{x+1}-\frac{2}{2x-5}-\frac{1}{x+4}}.}

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