If frac{d}{d x} left[ frac{(x+1)^2 sqrt{x-1}} | vedclass

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(C) Let $y = f(x) = \frac{(x+1)^2 \sqrt{x-1}}{(x+4)^3 e^x}$.Taking the natural logarithm on both sides:$\log y = \log \left[ \frac{(x+1)^2 \sqrt{x-1}}

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$\frac{8}{81 e^5}$

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(C) Let $y = f(x) = \frac{(x+1)^2 \sqrt{x-1}}{(x+4)^3 e^x}$.Taking the natural logarithm on both sides:$\log y = \log \left[ \frac{(x+1)^2 \sqrt{x-1}}{(x+4)^3 e^x} ight]$$\log y = 2 \log (x+1) + \frac{1}{2} \log (x-1) - 3 \log (x+4) - x$.Differentiating both sides with respect to $x$:$\frac{1}{y} \frac{d y}{d x} = \frac{2}{x+1} + \frac{1}{2(x-1)} - \frac{3}{x+4} - 1$.Thus,$\frac{d y}{d x} = y \left[ \frac{2}{x+1} + \frac{1}{2(x-1)} - \frac{3}{x+4} - 1 ight]$.Comparing this with the given equation,we identify $f(x) = y = \frac{(x+1)^2 \sqrt{x-1}}{(x+4)^3 e^x}$.Now,substitute $x = 5$ into $f(x)$:$f(5) = \frac{(5+1)^2 \sqrt{5-1}}{(5+4)^3 e^5} = \frac{6^2 \sqrt{4}}{9^3 e^5} = \frac{36 	imes 2}{729 e^5} = \frac{72}{729 e^5} = \frac{8}{81 e^5}$.

If $\frac{d}{d x} \left[ \frac{(x+1)^2 \sqrt{x-1}}{(x+4)^3 e^x} \right] = f(x) \left[ \frac{2}{x+1} + \frac{1}{2(x-1)} - \frac{3}{x+4} - 1 \right]$,then $f(5) = $

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