- #1

Imperial

For any of you who have done differential calculus, I need a little help with a problem involving natural logarithms.

The question asks to differentiate y = ln x from first principles . It says "use the definition of the Euler number, namely e = lim(n->inf.) (1+1/n)^n.".

First principles means f'(x) = lim(h->0) [f(x+h) - f(x)] / h (this is the first thing we learned in calculus).

I so far managed two different methods:

Method 1. y = ln x

therefore e^y = x

dx/dy = e^y.

Since dx/dy * dy/dx = 1

1/(dx/dy) = dy/dx.

= 1/e^y

= 1/x.

Method 2. y = ln x

f(x) = ln x

f(x+h) = ln (x+h)

f'(x) = lim(h->0) [ln (x+h) - ln x] / h

= lim(h->0) ln (x+h/x) / h

= lim(h->0) 1/h * ln(1+h/x)

Since lim(h->0) ln(1+h/x) -> h/x where h != 0,

f'(x) = 1/h * h/x

= 1/x

Both of these methods work and are valid, although I didn't bring the definition of the Euler number into it. I personally have no idea how to do this. Could anyone here who has done a bit of math before please help me with this?

Thanks.