Partial differentiation

When we take the partial derivative of $f(x,y)$ with respect to some variable (say $x$), then we hold the other variables ($y$) as constant. Geometrically, the partial derivative is like a trace at some constant value (for $f_x$, it is constant $y$) of the derivative function.

$$\frac{\partial }{\partial x}f(a,b)=\underset{h\to 0}{\lim }\frac{f(a+h,b)-f(a,b)}{h}$$

By theorem (the Clairaut equality of mixed partial derivatives), if $f$ is defined on an open set $D$ of ${\mathbb{R}}^2$, and $f_{xy}$ and $f_{yx}$ are continuous throughout $D$, then $f_{xy}$ and $f_{yx}$ at all points of $D$. This holds for higher derivatives of $f$. For example, if all are continuous, then $f_{xyx}=f_{xxy}=f_{yxx}$.

The notation can get pretty confusing:

At some point

Sometimes when we’re asked to evaluate the partial derivative at some point $(a,b)$, it makes sense to use the definition and plug $(0,0)$ in, instead of computing what the derivative is, then evaluating at that point.

See also

• Directional derivative
• Gradient
• Tangent plane