Answer:
[tex]\frac{dy}{dx}=-[(\frac{5x+24}{36x-6x^2})][/tex]
Step-by-step explanation:
Given function:
y =[tex]\ln(\frac{(6 - x)^{\frac{3}{2}}}{x^{\frac{2}{3}}})[/tex]
we know
[tex]\ln(\frac{A}{B})[/tex] = ln(A) - ln(B)
thus,
y = [tex]\ln((6 - x)^{\frac{3}{2}}) - \ln(x^{\frac{2}{3}})[/tex]
or
also,
ln(Aⁿ) = n × ln(A)
thus,
y = [tex](\frac{3}{2})\times\ln(6 - x) - (\frac{2}{3})\times\ln(x)[/tex]
therefore,
[tex]\frac{dy}{dx}=[(\frac{3}{2})\times\frac{1}{(6-x)}\times(0 - 1)] - [ (\frac{2}{3})\times\frac{1}{x}\times1][/tex]
or
[tex]\frac{dy}{dx}=-(\frac{3}{2(6-x)}) - (\frac{2}{3x})[/tex]
or
[tex]\frac{dy}{dx}=-[(\frac{3(3x)+2\times2(6-x)}{2(6-x)\times(3x)})][/tex]
or
[tex]\frac{dy}{dx}=-[(\frac{5x+24}{36x-6x^2})][/tex]
