\(\begin{array}{l}
\quad f(x,y,z) = e^{\dfrac{1}{x^2 + y^2 +z^2}}\\
\bullet\quad \dfrac{\partial f}{\partial x} = \left(\dfrac{1}{x^2 + y^2 + z^2}\right)'\cdot e^{\dfrac{1}{x^2 + y^2 +z^2}}\\
\Leftrightarrow \dfrac{\partial f}{\partial x} = - \dfrac{2x}{(x^2 + y^2 + z^2)^2}\cdot e^{\dfrac{1}{x^2 + y^2 +z^2}}\\
\bullet\quad \dfrac{\partial f}{\partial x} = \left(\dfrac{1}{x^2 + y^2 + z^2}\right)'\cdot e^{\dfrac{1}{x^2 + y^2 +z^2}}\\
\Leftrightarrow \dfrac{\partial f}{\partial y} = - \dfrac{2y}{(x^2 + y^2 + z^2)^2}\cdot e^{\dfrac{1}{x^2 + y^2 +z^2}}\\
\bullet\quad \dfrac{\partial f}{\partial x} = \left(\dfrac{1}{x^2 + y^2 + z^2}\right)'\cdot e^{\dfrac{1}{x^2 + y^2 +z^2}}\\
\Leftrightarrow \dfrac{\partial f}{\partial z} = - \dfrac{2z}{(x^2 + y^2 + z^2)^2}\cdot e^{\dfrac{1}{x^2 + y^2 +z^2}}\\
\bullet\quad df = \dfrac{\partial f}{\partial x}dx + \dfrac{\partial f}{\partial y}dy + \dfrac{\partial f}{\partial z}dz\\
\Leftrightarrow df = - \dfrac{2x}{(x^2 + y^2 + z^2)^2}\cdot e^{\dfrac{1}{x^2 + y^2 +z^2}}dx\\
\kern37pt - \dfrac{2y}{(x^2 + y^2 + z^2)^2}\cdot e^{\dfrac{1}{x^2 + y^2 +z^2}}dy\\
\kern37pt- \dfrac{2z}{(x^2 + y^2 + z^2)^2}\cdot e^{\dfrac{1}{x^2 + y^2 +z^2}}dz
\end{array}\)
