If is a derivative of at , then . In particular, if is differentiable at , these partial derivatives exist and the derivative is unique.

Proof:

Let , then becomes since .

By choosing (all zeroes except in th position), then as , and So .

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# Multivariable Derivative and Partial Derivatives

If is a derivative of at , then . In particular, if is differentiable at , these partial derivatives exist and the derivative is unique.

Proof:

Let , then becomes since .

By choosing (all zeroes except in th position), then as , and So .

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