Show that the derivative of ln(x) = 1/x

We can start by letting y = ln(x)

What we are trying to show is that dy/dx = 1/x

Since y = ln(x), then e^y = e^ln(x) = x

Taking the derivative of each side of this equation will give us e^y.dy/dx = 1

If we divide each side of this new equation by e^y then we have that dy/dx = 1/e^y = 1/x as required.