I was recently introduced to the wave equation and in the derivation of said equation they used the definition of the second order partial derivative for the final result. The source I used for the definition basically said that the following equation:
is the equivalent of:
And mentioned that this is because the partial second order derivative of u w.r.t x is defined as such.
I presumed that this meant if derived from first principles (taking lim h->0) for the definied formula one gets the second order partial derivative. However, I could not find an online resource which derives such a formula. I have tried to attempt this myself, however, I do not understand where the h squared term comes from.
If someone knows of the derivation, if they could kindly direct me to it, that would be much appreciated. Thanks.
Best Answer
I think the insight can be seen just by seeing single variable derivatives:
$$ f' = \lim_{h \to 0} \frac{ f(x+h) - f(x) }{h}$$
$$ f'' = \lim_{h \to 0} \frac{ f'(x+h) - f'(x) }{h}= \lim_{h \to 0} \frac{ \frac{ f(x+2h) - f(x+h) }{h} - \frac{ f(x+h) - f(x) }{h} }{h}$$
Or,
$$ f'' = \lim_{h \to 0} \frac{ f(x+2h) + f(x) - 2 f(x+h)}{h^2}$$