I have mostly been using the derivative's limit definition, hence the second derivative and higher-order derivatives. However, today I came across this strange relation for the second derivative in one of the texts. Can someone help me understand how we got this relation for the second derivative on the right-hand side from the difference of the first derivatives on the left-hand side? $$ \left.\frac{d T}{d x}\right|_{x+\frac{\Delta x}{2}}-\left.\frac{d T}{d x}\right|_{x-\frac{\Delta x}{2}} = \Delta x \frac{d^2T}{dx^2} $$
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1This is not a definition of the second derivative. What text book are you reading? – Kavi Rama Murthy Jun 20 '21 at 07:23
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Sorry, this is not the definition of the second derivative for sure, which I will change now. But, they used this relation without giving any context to it. – Roshan Shrestha Jun 20 '21 at 07:25
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It is also not a valid equation. – Kavi Rama Murthy Jun 20 '21 at 07:28
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Sorry, I missed adding $\Delta x$ on the right-hand side. – Roshan Shrestha Jun 20 '21 at 07:30
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2It helps to first understand that $f'(x) \approx \frac{f(x + \Delta x/2) - f(x - \Delta x/2)}{\Delta x}$. – littleO Jun 20 '21 at 07:31
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@littleO Thank you very much. Got it :-) – Roshan Shrestha Jun 20 '21 at 07:49