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Example Find the total differential of the following function.

$\displaystyle f(x,y) = \log(x^2 + y^2) $

Answer The total differential of $z = f(x,y)$ is that for the surface $z = f(x,y)$, the effect in the $z$ direction caused by a small change int the $x$ direction and the effect in the $z$ direction caused by a small change int the $y$ direction．When you put these effects together, we have the total differential.

$\displaystyle df = f_{x}(x,y)dx + f_{y}(x,y)dy$

We denote $(f_{x}(x,y),f_{y}(x,y)$ by $\nabla f(x,y)$. Then the total differential of $f(x,y)$ is given by

$\displaystyle df = \nabla f(x,y) \cdot (dx, dy)$

Now we solve the problem.

$\displaystyle f_{x}(x,y) = \frac{2x}{x^2 + y^2}, f_{y} = \frac{2y}{x^2 + y^2}$

Thus，

$\displaystyle df = \frac{2x}{x^2 + y^2}dx + \frac{2y}{x^2 + y^2}dy$