Question

What are the first and second derivatives of `f(x)=x^8*ln(x)`?

Answer 1

Let's remember product rule: be `y=f(x)g(x)`, then `y'=f'(x)g(x)+f(x)g'(x)`

Explanation:

Solving:

`(df(x))/(dx)=8x^7lnx+x^8(1/x)=8x^7lnx+x^7=x^7(8lnx+1)`

Using product rule again:

`(df(x))^2/(d^2x)=7x^6(8lnx+1)+x^7(8/x)`

`(df(x))^2/(d^2x)=56x^6lnx+7x^6+8x^6=56x^6lnx+15x^6`

`(df(x))^2/(d^2x)=x^6(56lnx+15)`