Question

Using first principle find derivative of log ax+b

Answer 1

Solution:

Let y=log(ax+b), then y+dy=log(a(x+dx)+b), where dy is the small change in y for a small change dx in x,

Then dy=y+dy-dy
=log(a(x+dx)+b)-log(ax+b)
=log((ax+adx+b)/(ax+b))
=log(1+adx/(ax+b)).

Natural logs are assumed where log(1+x)=x, to the first approximation.
So dy=adx/(ax+b) and dy/dx=a/(ax+b) or a/y.

(Answer)