Question

In Lecture 6, Mr. David Jerison talks about natural logs, and presents to us:
w=ln(x) so e^w=x, (d/dx)e^w=(d/dx)x=1, so [(d/dx)e^w](dw/dx)=1
This implies (dw/dx)=1. How so, and how did he even arrive at this conclusion?

Answer 1

You missed a step. He simplified the (d/dx)e^w to simply e^w. It went something like this:
w=lnx
e^w=x
(d/dx)e^w=(d/dx)x
So here we know we have to chain rule the e^w.
e^w(dw/dx)=1
(dw/dx)=1/(e^w)
Just substitute back the x
(dw/dx)=1/x

Answer 2

The way he wrote it was a bit weird for me too.

Answer 3

Oh so is it kind of like
\[(dx/dx)e^w*(dw/dx) \]
and the (dx/dx) cancels out?