Find the derivative 1. g(x)= x/ln x?

Answer 1

The quotient rule is the easiest to use for this one.

In a fraction such that, f(x)/g(x) is given. Use this rule:

( f'(x)g(x) - g'(x)f(x) ) / g(x)^2

g'(x) = (1)(ln x) - (1/x)(x) / (ln x)^2

g'(x) = (ln x - 1) / (ln x)^2

That is the first derivative.

Answer 2

Quotient rule
g `(x) = [ (ln x).1 - x.(1/x) ] / (ln x)²
g `(x) = [ ln x - 1 ] / ( ln x)²

Answer 3

Use the product rule: x(ln(x))^-1, then the chain rule
g = xy^-1, where y = ln (x)
g' = y^-1dx - xy^-2 dy
dy = dx/x
g' = y^-1dx -y^-2dx
= dx[(1/y) - (1/y^2)]
g' = [ln(x) - 1]/(ln(x)^2)