$\displaystyle f(x;y)=\sin(x^2y^3z^4).$

 

Дифференциал функции будем находить по формуле

 

$\displaystyle df=df(x,y,z;dx,dy,dz)=\frac{\partial f}{\partial x}dx+
\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz.$

Поскольку

 

$\displaystyle \frac{\partial f}{\partial x}=\cos(x^2y^3z^4)\cdot(x^2y^3z^4)'_x=
\cos(x^2y^3z^4)\cdot2xy^3z^4,$

 

$\displaystyle \frac{\partial f}{\partial y}=\cos(x^2y^3z^4)\cdot(x^2y^3z^4)'_y=
\cos(x^2y^3z^4)\cdot3x^2y^2z^4$

и

 

$\displaystyle \frac{\partial f}{\partial z}=\cos(x^2y^3z^4)\cdot(x^2y^3z^4)'_z=
\cos(x^2y^3z^4)\cdot4x^2y^3z^3,$

получаем

 

$\displaystyle df=\cos(x^2y^3z^4)\Bigl(2xy^3z^4dx+3x^2y^2z^4dy+4x^2y^3z^3dz\Bigr).$