Question

What are the first and second derivatives of `f(x)=-3x^5lnx`?

Answer 1

Here, we'll need the product rule, which states that for `y=f(x)g(x)`, `y'=f'(x)g(x)+f(x)g'(x)`

Explanation:

Solving:

`(df(x))/(dx)=(-15x^4)(lnx)+(-3x^5)(1/x)=-15x^4lnx-3x^4=color(green)(-3x^4(5lnx+1))`

Second derivative follows the same logic:

`(df(x))^2/(dx)=(-12x^3)(5lnx+1)+(-3x^4)(5/x)=-60x^3lnx-12-15x^3`

`(df(x))^2/(dx)=color(blue)(-15x^3(4lnx+1)-12)`