Question

How do you differentiate `f(x) = [ (2x)/(log(x)) ] - [ x/(log²(x)) ]`?

Answer 1

Refer to explanation

Explanation:

First we find the derivative of `(2x)/logx` hence we have that

`((2x)/logx)'=((2x)'*logx-2x(logx)')/(logx)^2=(2*logx-2)/(logx)^2`

and then the derivative of `x/(logx)^2` hence we have that

#(x/(logx)^2)'=((x)'*(logx)^2-x((logx)^2)')/(logx)^4= ((logx)^2-2xlogx*(1/x))/(logx)^4=((logx-2))/(logx)^3#

Finally we get

`f'(x)=(2*(logx)^2-3logx+2)/(logx)^3`

Remarks

1).The derivative of a quotient function is

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

2).The derivative of logx is `(logx)'=1/x`