Partial Differentiation with Respect to One Variable

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.

Example
If r = cos (xy) + 3xy – 2x2 – 3x – 2y, find ∂F/∂x and ∂F/∂y.

Solution:

Partial Differentiation with Respect to Several Variables

The partial derivative of F with respect to x then to y is

$\dfrac{\partial^2 F}{\partial x \, \partial y} = \dfrac{\partial}{\partial y} \left( \dfrac{\partial F}{\partial x} \right)$

The partial derivative of F with respect to y then to x is

$\dfrac{\partial^2 F}{\partial y \, \partial x} = \dfrac{\partial}{\partial x} \left( \dfrac{\partial F}{\partial y} \right)$

Note:

$\dfrac{\partial^2 F}{\partial x \, \partial y} = \dfrac{\partial^2 F}{\partial y \, \partial x}$

Example
Given F = sin (xy). Show that ∂2F / (∂x ∂y) is equal to ∂2F / (∂y ∂x).

Solution:

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