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space such that

where · is the dot product.

As a consequence, the usual properties of the derivative hold for the gradient, though the gradient is not a derivative itself, but rather dual to the derivative:

;Linearity
:The gradient is linear in the sense that if and are two real-valued functions differentiable at the point , and and are two constants, then is differentiable at , and moreover ;Product rule
:If and are real-valued

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