Why is the derivative of ln(x) equal to 1/x.

This fact can be deduced several ways, and I would encourage tutees to find other methods, (such as using implicit differentiation, or the chain rule.) However one of the simplest explanations is given below.To start, we know that due to the properties of the irrational number e, (and in fact, it is one of the ways e can be defined!) the derivative of e^x is also e^x. Now, to calculate the derivative of ln(x), consider the equation: y=ln(x) (1)Raising both sides as powers of e:e^y=e^ln(x) (2)And using our log rules gives us:e^y=x, (3)and rearranging gives us:x=e^y (4)Now x and y and both just variables, so we can differentiate with respect to whatever we choose, and so, differentiating with respect to y (rather than x) shows us:dx/dy=e^yAs expected. Now we can exploit a fact about derivatives, which is that dy/dx= 1/(dx/dy) (which can easily be confirmed by considering the definition of a derivative, as Δy/Δx=1/(Δx/Δy) ). Thus with some more rearrangement: 1/(dx/dy)=1/e^y (5)and thus:dy/dx=1/e^y (6)Now this result isn't quite what we were looking for, as we were hoping to have the derivative be a function of x. However we can see from equation (3) that e^y=x, so with a final substitution we can see that:dy/dx=1/xWhich is the result we were looking to show!