OK  a spot of A-level maths comes to the blog

A few things to remember:

1. Ln x – The Natural log of x – is the inverse function ^{Footnote 1} of e^x
2. d/dx e^x = e^x  – That’s what so special about ex

3. Remember the result from last time that

 

 

 

 

Now how can we use these three facts to find d/dx ln x?

Let f(x) = y = Ln x . We want f’(x) = dy/dx ^{Footnote 2}
Raise e to both sides

e^y = e^{ln x}  = x  (Using 1 above)

so  x = e^y  so dx/dy = e^y  (Using 2 above)

so dy/dx = 1/ e^y

e^y = x so substitute this in the give f’(x) in terms of x and we get 1/x.

 

d/dx  ln x = 1/x.

 

Footnotes

1 I use the word function here but caution is required. You must be careful with the domain of Ln x if it’s going to meet the criteria of a Function – It must be defined as x≥0

2. Ok so I’m mixing my calculus formats here! Both have their uses, maybe the subject for another blog entry.