Question

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

y = (ln x2)2

Answer 1

Answer: The required derivative is
`(dy)/(dx)=\\frac{4}{x\ln x^2}`

Explanation:

Since we have given that

`y=(\ln x^2)^2`

We need to derivative it w.r.t 'x'., using "Chain rule"

As we know that

`(d)/(dx) \ln x=(1)/(x)\\\\and\\\\(d)/(dx)x^n=nx^(n-1)`

So, it becomes,

`(dy)/(dx)=(1)/(\ln (x^2)^2)* 2\ln(x^2)* (1)/(x^2)* 2x\\\\(dy)/(dx)=(4)/(x\ln x^2)`

Hence, the required derivative is
`(dy)/(dx)=\\frac{4}{x\ln x^2}`

Answer 2

`(dy)/(dx)=(8lnx)/(x)`

Explanation:

We are given that a function

`y=(lnx^2)^2`

We have to find the derivative of the function

Differentiate w.r.t x

`(dy)/(dx)=2(lnx^2)* (1)/(x^2)* 2x`

By using formula

`(d(lnx))/(dx)=(1)/(x)`

`(dx^n)/(dx)=nx^(n-1)`

`(dy)/(dx)=(4lnx^2)/(x)=(4(2)lnx)/(x)`

by using
`lnx^y=ylnx`

Hence, the derivative of function

`(dy)/(dx)==(8lnx)/(x)`