What is the derivative of ln(x)?

First let y=ln(x).

Recall that the exponential function, e^x, is defined as the inverse of the logarithmic function, ln(x).

To make x the subject of the formula, use the inverse function exp. This gives that x=e^y.

Now, differentiate both sides with respect to y and recall that d/dx(e^x)=e^x. This gives dx/dy=e^y.

Remember for a derivative, dy/dx=1/(dx/dy).

Therefore, dy/dx=1/e^y.

Finally, from above, x=e^y.

Substituting for e^y we have dy/dx=1/x which is our final result.

Therefore the derivative of ln(x), is dy/dx=1/x.