# partial differentiation

## Partial Derivatives

Let F be a function of several variables, say x, y, and z. In symbols,

$F = f(x, \, y, \, z)$.

The partial derivative of F with respect to x is denoted by

$\dfrac{\partial F}{\partial x}$

and can be found by differentiating f(x, y, z) in terms of x and treating the variables y and z as constants.

## The Determination of Integrating Factor

From the differential equation

$M ~ dx + N ~ dy = 0$

**Rule 1**

If $\dfrac{1}{N}\left( \dfrac{\partial M}{\partial y} - \dfrac{\partial N}{\partial x} \right) = f(x)$, a function of

*x*alone, then $u = e^{\int f(x)~dx}$ is the integrating factor.

**Rule 2**

If $\dfrac{1}{M}\left( \dfrac{\partial M}{\partial y} - \dfrac{\partial N}{\partial x} \right) = f(y)$, a function of

*y*alone, then $u = e^{-\int f(y)~dy}$ is the integrating factor.