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Partial differentiation

In Chapters 3 and 4 we discussed functions *f* of only one variable *x*, which were usually written *f*(*x*). Certain constants and parameters may also have appeared in the definition of *f*, e.g. *f*(*x*) = *ax* + 2 contains the constant 2 and the parameter *a*, but only *x* was considered as a variable and only the derivatives *f*^{(n)}(*x*) = *d ^{n}f*/

However, we can equally well consider functions that depend on more than one variable, e.g. the function *f*(*x, y*) = *x*^{2} + 3*xy*, which depends on the two variables *x* and *y*. For any pair of values *x, y*, the function *f*(*x, y*) has a well-defined value, e.g. *f*(2*,* 3) = 22. This notion can clearly be extended to functions dependent on more than two variables. For the *n*-variable case, we write ...

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