I wish to prove the following.

"Let be represented by Taylor series at , where , and let the first non-vanishing derivatives at be and , where . Then

"

So here's how I've proceeded so far:

If we let , then

Differentiating wrt :

Again, letting , then

And so on, so that:

and similarly,

So:

Recalling that and are the first non-vanishing derivatives at . Then:

Recalling that , then:

But unless , as , and so , which clearly is not the desired result.

Where am I going wrong?