Question

Finding Higher-Order Derivatives In Exercise, find the second derivative of the function.

f(x) = 3 + 2 ln x

Answer 1

`f''(x)=-(2)/(x^2)`

Explanation:

We are given that

`f(x)=3+2lnx`

We have to find the second order of given function

To find the highest order derivative using the formula given below

`(d^nA)/(dx^n)=(d((d^(n-1)A)/(dx^(n-1))))/(dx)`

Differentiate w.r.t.x

`f'(x)=2(1)/(x)=(2)/(x)=2x^(-1)`

By using the formula
`(d(lnx))/(dx)=(1)/(x)`

`(1)/(x)=x^(-1)`

Substitute n=2

`f''(x)=(d^2f(x))/(dx^2)=(d(f'(x)))/(dx)=-2x^(-2)=-(2)/(x^2)`

By using the formula

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

Hence, the second order derivative of function is given by

`f''(x)=-(2)/(x^2)`