## Partial Differentiation

by AppliedMathematics.info

Printable Worksheet: Partial Differentiation

Partial differentiation has meaning in the context of the differentiation of functions of several variables. For example let *f*(*x*,*y*,*z*) be a function of *x*,*y*,*z*. The first order partial derivative of *f* with respect to *x* is deined as

| ¶*f* ¶*x*
| = | lim *h*® 0
| | *f*(*x*+*h*,*y*,*z*)-*f*(*x*,*y*,*z*) *h*
| , | |

whenever the limit exists. The partial derivatives of *f* with respect to *y* and *z* are defined similarly. The partial derivatives can be written *f*_{x}, *f*_{y}, *f*_{z}. By differentiating these functions again with respect to *x*, *y*, or *z* we obtain the higher order partial derivatives such as

| ¶^{2} *f* ¶*x*^{2}
| , | ¶^{2} *f* ¶*x*¶*y*
| . | |

__Chain Rule__

Let *F* be a continuously differentiable function of *f*,*g*,*h* and suppose that each *f*,*g*,*h* is a continuously differentiable function of *x*,*y*,*z*. Then it may be shown that *F* is a continuously differentiable function of *x*,*y*,*z* and that

| ¶*F* ¶*x*
| = | ¶*F* ¶*f*
| | ¶*f* ¶*x*
| + | ¶*F* ¶*g*
| | ¶*g* ¶*x*
| + | ¶*F* ¶*h*
| | ¶*h* ¶*x*
| , | |

| ¶*F* ¶*y*
| = | ¶*F* ¶*f*
| | ¶*f* ¶*y*
| + | ¶*F* ¶*g*
| | ¶*g* ¶*y*
| + | ¶*F* ¶*h*
| | ¶*h* ¶*y*
| , | |

| ¶*F* ¶*z*
| = | ¶*F* ¶*f*
| | ¶*f* ¶*z*
| + | ¶*F* ¶*g*
| | ¶*g* ¶*z*
| + | ¶*F* ¶*h*
| | ¶*h* ¶*z*
| , | |

An equation formed from partial derivatives is called a partial differential equation .

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