Question

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

y = ln[x(2x + 3)2]

Answer 1

Answer: The required derivative is
`(8x^2+18x+9)/(x(2x+3)^2)`

Explanation:

Since we have given that

`y=\ln[x(2x+3)^2]`

Differentiating log function w.r.t. x, we get that

`(dy)/(dx)=(1)/([x(2x+3)^2])* [x'(2x+3)^2+(2x+3)^2'x]\\\\(dy)/(dx)=(1)/([x(2x+3)^2])* [(2x+3)^2+2x(2x+3)]\\\\(dy)/(dx)=(4x^2+9+12x+4x^2+6x)/(x(2x+3)^2)\\\\(dy)/(dx)=(8x^2+18x+9)/(x(2x+3)^2)`

Hence, the required derivative is
`(8x^2+18x+9)/(x(2x+3)^2)`