Question

If y= logx/x find yn in lebinetz theorem

Answer 1

Answer: First we need to do a change of base from 10 to e.

log x = ln x / ln 10 = (1 / ln 10) ln x

Then as you start taking derivatives, the (1 / ln 10) will just be a constant.

The first derivative of ln x is 1/x

The second derivative of ln x is -1/(x^2)

The third derivative of ln x is 2 /(x^3)

The fourth derivative of ln x is -6/(x^4)

See the pattern? The derivatives alternate between positive and negative. We will accomplish this with the standard (-1)^(n+1). The value of the coefficient is (n-1)! This is all times (1 / x^n). And do not forget the constant ( 1 / ln 10 )

So the nth derivative of log x is ( 1 / ln 10) ((-1)^(n+1)) (n-1)! (1 / x^n)

Fun!!