Find the derivative of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y\,=\,\ln {\frac {(x+1)^{4}}{(2x-5)(x+4)}}.}
| Foundations:
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| This problem is best approached through properties of logarithms. Remember that
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Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \ln(xy)=\ln x+\ln y,}
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| while
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| and
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| 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 \ln \left( x^n \right) = n\ln x,}
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| You will also need to apply
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| 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
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| 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).}
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| Finally, recall that the derivative of natural log is
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- 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}.}
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Solution:
| Step 1:
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| We can use the log rules to rewrite our function as
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- 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}}
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| Step 2:
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| We can differentiate term-by-term, applying the chain rule to each term to find
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- 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}}
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| Final Answer:
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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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