Partial Derivatives

Partial derivatives are derivatives of functions of more variable are those which are taken with respect to one variable and holding all others constant, ignoring any possible interdependencies that the chain rule would normally account for. The partial derivative is represented by the symbol `\partial`.

The partial derivative of a function `f` with respect to `x` can be written as `\dfrac\partial{\partial x}f`, `\dfrac{\partial f}{\partial x}`, `\partial_xf`, or even `f_x`.

For instance, given a function `f(x,y)=x^2y+3x+y-10`, the partial derivative with respect to `x` would be `\dfrac{\partial f}{\partial x}=2xy+3`.
Taken with respect to `y`, the partial derivative would be `\dfrac{\partial f}{\partial y}=x^2+1`

There is an implicit difference in the significance of the partial derivative notation versus that of normal Leibniz notation for derivatives.
With traditional single-variable (or total) derivatives, they can often be treated like fractions as a result of the chain rule and similar manipulations. ^{[src 1] [src 2]}
These tricks often fail when extended to partial derivatives, and should be avoided. The partial derivative should be viewed more as an operator than a fraction in every circumstance.

Important Note: Differentials with functions of multiple variables are quite logical. For instance, given a function `z(x,y)`, the total differential is
`\mathrm dz=\mathrm dx\dfrac{\partial z}{\partial x}+\mathrm dy\dfrac{\partial z}{\partial y}` and the total derivative is `\dfrac{\mathrm dz}{\mathrm dt}=\dfrac{\mathrm dx}{\mathrm dt}\dfrac{\partial z}{\partial x}+\dfrac{\mathrm dy}{\mathrm dt}\dfrac{\partial z}{\partial y}`.

Important Note: Integrals are also different when dealing with partial derivatives. Normally, there is a constant of integration with single-variable functions, but now, except for with total derivatives or gradients, there is an arbitrary function of one or more variables left over. For instance,
`\displaystyle\int\frac{\mathrm dy}{\mathrm dx}\,\mathrm dx=y+C`, but `\displaystyle\int\frac{\partial}{\partial x}f(x,y)\,\mathrm dx=f(x,y)+C(y)` ^[src]

Important Note: `f_{xy}=f_{yx}` holds true for every function `f`. In other words, the order of partial differentiation is inconsequential.