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How to show this:

$f(x,y+en,y'+en')-f(x,y,y')= en(df/dy)+e(dn/dx)(df/dy')+O(e^2)$

y and n are functions of x, e small constant And y is smooth.

What identities or properties are used here?

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up vote 0 down vote accepted

You did not give any assumptions on $f$. It suffices to assume that the first-order partial derivatives of $f$ are continuous, and second-order derivatives are bounded (in a neighborhood of the point of interest). Then the following form of multivariable Taylor expansion holds: $$ f(u,v,w) = f(u_0,v_0,w_0) + \frac{\partial f}{\partial u}(u_0,v_0,w_0)\,(u-u_0) + \frac{\partial f}{\partial v}(u_0,v_0,w_0)\,(v-v_0) + \frac{\partial f}{\partial w}(u_0,v_0,w_0)\,(w-w_0) +O\left((u-u_0)^2+(v-v_0)^2+(w-w_0)^2\right) $$ In your situation $u=u_0=x$, $v_0=y$, $v=y+e n$, $w_0=y'$, $w=y'+en' = y'+e(dn/dx)$. This yields the identity.

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