Question

Differentiating a Logarithmic Function in Exercise, find the derivative of the function. See Examples 1, 2, 3, and 4.

Answer 1

Answer:

$\frac{dy}{dx}=-[(\frac{5x+24}{36x-6x^2})]$

Step-by-step explanation:

Given function:

y =$\ln(\frac{(6 - x)^{\frac{3}{2}}}{x^{\frac{2}{3}}})$

we know

$\ln(\frac{A}{B})$ = ln(A) - ln(B)

thus,

y = $\ln((6 - x)^{\frac{3}{2}}) - \ln(x^{\frac{2}{3}})$

or

also,

ln(Aⁿ) = n × ln(A)

thus,

y = $(\frac{3}{2})\times\ln(6 - x) - (\frac{2}{3})\times\ln(x)$

therefore,

$\frac{dy}{dx}=[(\frac{3}{2})\times\frac{1}{(6-x)}\times(0 - 1)] - [ (\frac{2}{3})\times\frac{1}{x}\times1]$

or

$\frac{dy}{dx}=-(\frac{3}{2(6-x)}) - (\frac{2}{3x})$

or

$\frac{dy}{dx}=-[(\frac{3(3x)+2\times2(6-x)}{2(6-x)\times(3x)})]$

or

$\frac{dy}{dx}=-[(\frac{5x+24}{36x-6x^2})]$