Writing higher order derivatives using the limit definition of derivative?

Question

Studying Taylor series, I wanted to get a sense for what higher derivatives really express in precise terms using the limit definition of the derivative.

Is this correct?

$$\frac{d^2y}{dx^2} = \lim_{h \to 0} \frac{\big(\lim_{k \to 0} \frac{f(x+h+k)-f(x+h)}{k}\big) - \big(\lim_{g \to 0} \frac{f(x+g)-f(x)}{g}\big)}{h}$$

Answer 1

You are right, but you could do with an easier expression in one limit: $$ f''(x) = \lim_{h \to 0} \frac{f(x+2h) - 2f(x+h) + f(x)}{h^2} $$