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Differentiation of the Ln function.

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 ex
  2. d/dx ex = ex  – 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

ey = eln x  = x  (Using 1 above)

so  x = ey  so dx/dy = ey  (Using 2 above)

so dy/dx = 1/ ey

ey = 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.

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