OK a spot of A-level maths comes to the blog
A few things to remember:
- Ln x – The Natural log of x – is the inverse function Footnote 1 of ex
- d/dx ex = ex – That’s what so special about ex
- 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.
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.