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I tried to derive the formula second derivatives from Wikipedia:

$$f''\left( x\right) =\lim _{h\rightarrow 0}\dfrac {f\left( x+h\right) -2f\left( x\right) +f\left( x-h\right) }{h^{2}}$$

Let $y=f(u)$ and using first principles, since $$\dfrac {dy}{du}=\lim _{k\rightarrow 0}\dfrac {f\left( u+k\right) -f\left( u\right) }{k}$$ therefore, $$\dfrac {d^{2}y}{dx^{2}}=\dfrac {d\left( \dfrac {dy}{du}\right) }{dx}= \lim _{h\rightarrow 0} \dfrac{\left.\dfrac {dy}{du}\right|_{u= x+h}-\left.\dfrac {dy}{du}\right|_{u=x}}{h}\\ = \lim _{h\rightarrow 0}\dfrac {\lim _{k\rightarrow 0}\dfrac {f\left( x+h+k\right) -f\left( x+h\right) }{k}-\lim _{k\rightarrow 0}\dfrac {f\left( x+k\right) -f\left( x\right) }{k}}{h} \\ = \lim _{h\rightarrow 0}\dfrac {\lim _{k\rightarrow 0}\dfrac {f\left( x+h+k\right) -f\left( x+h\right) -f\left( x+k\right) +f\left( x\right) }{k}}{h} $$ And I’m stuck with a stacked limit of two bound variables. So what have I done wrong?

gldanoob
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    You may be stuck,but you have done nothing wrong. To get this formula you need more hypotheses than just the existence of $f''$ at the point $x$. It's usually done by a L'Hospital's Rule argument (taking care that the applications are valid!) – ancient mathematician Dec 23 '19 at 15:55
  • @ancientmathematician could you show that where exactly should I apply the LH rule? Thanks – gldanoob Dec 24 '19 at 00:16
  • Apply L'Hospital's Rule (only once) to the expression under limit. Remember that you have to differentiate both numerator and denominator with respect to the limit variable $h$ and not $x$. – Paramanand Singh Dec 24 '19 at 03:59
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    I'm not inclined to write out what is In The Books. But you need that $f'$ exists in a neighbourhood of $x$ and that $f''$ exists at $x$. Then apply LHR to the RHS, and get that it equals $\lim_{h\to 0} (f'(x+h)-f'(x-h)/2h$. Add $-f'(x)+f'(x)$ to numerator, separate into two parts and use definition of $f''(x)$ twice and you're done. – ancient mathematician Dec 24 '19 at 07:31
  • Why not apply L’Hospitals rule a second time to get the answer! – Lawrence Mano Apr 26 '23 at 06:12

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