What is the derivative of ##ln(1+(1/x))##?

##d/dx(ln(1+(1/x))) = (-1)/(x(x+1))##

Although you could use ##d/dx (ln(u)) = 1/u (du)/dx##, the algebra will get messy that way.

Let’s rewrite using properties of ##ln##.

##y = ln(1+(1/x)) = ln((x+1)/x)##

## = ln(x+1) – ln(x)##

So

##dy/dx = 1/(x+1) – 1/x = (-1)/(x(x+1))##