Изменения

<tex> f(x) - f(x_0) = \sum \limits_{k=1}^n \frac {f^{(k)}(x_0)}{k!}(x-x_0)^k+\frac {f^{(n+1)}(x_0+\theta (x-x_0))}{(n+1)!}(x-x_0)^{n+1} </tex>

<tex>d^k f(x_0)=f^{(k)}(x_0)\Delta x^k</tex>

<tex>g(t)=f(t,y+\Delta y)-f(t,y)</tex>

<tex>g'(t)=\frac {\partial f}{\partial x}(t,y+\Delta y)-\frac {\partial f}{\partial x}(t,y)</tex>

<tex>d^2 f=d\left( \frac {\partial f}{\partial x}(\overline x) \Delta x + \frac {\partial f}{\partial y}(\overline x) \Delta y\right)</tex><tex>=\frac{\partial^2 f}{\partial x^2}(\overline x) \Delta x^2+2\frac{\partial^2f}{\partial x \partial y}(\overline x) \Delta x\Delta y+\frac{\partial^2 f}{\partial y^2}(\overline x) \Delta y^2</tex>. Частные производные — непрерывны. Теперь пусть <tex>dx=\Delta x</tex>, <tex>dy=\Delta y</tex>: <tex>x=a+bt</tex>, <tex>dx=bdt</tex>