Question

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

y = ln{[(6 - x)^3/2]/x^2/3}

Answer 1

`(dy)/(dx)=-[((5x+24)/(36x-6x^2))]`

Explanation:

Given function:

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

we know

`\ln((A)/(B))` = ln(A) - ln(B)

thus,

y =
`\ln((6 - x)^{(3)/(2)}) - \ln(x^{(2)/(3)})`

or

also,

ln(Aⁿ) = n × ln(A)

thus,

y =
`((3)/(2))*\ln(6 - x) - ((2)/(3))*\ln(x)`

therefore,

`(dy)/(dx)=[((3)/(2))*(1)/((6-x))*(0 - 1)] - [ ((2)/(3))*(1)/(x)*1]`

or

`(dy)/(dx)=-((3)/(2(6-x))) - ((2)/(3x))`

or

`(dy)/(dx)=-[((3(3x)+2*2(6-x))/(2(6-x)*(3x)))]`

or

`(dy)/(dx)=-[((5x+24)/(36x-6x^2))]`