Product rule for partial differentiation
From Calculus
Statement for two functions
Statement for partial derivatives for functions of two variables
The derivatives used here are partial derivatives.
Version type | Statement |
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specific point, named functions | Suppose are both functions of variables . Suppose is a point in the domain of both and . Suppose the partial derivatives and both exist. Let denote the product of the functions. Then, we have: Suppose the partial derivatives and both exist. Then, we have: |
generic point, named functions | Suppose are both functions of variables . These hold wherever the right side expressions make sense (see concept of equality conditional to existence of one side). |
generic point, named functions, point-free notation | Suppose are both functions of variables . These hold wherever the right side expressions make sense (see concept of equality conditional to existence of one side). |
Statement for partial derivatives for functions of multiple variables
Version type | Statement |
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specific point, named functions | Suppose are both functions of variables . Suppose is a point in the domain of both and . Fix a number in . Suppose the partial derivatives and both exist. Let denote the product of the functions. Then, we have: |
generic point, named functions | Suppose are both functions of variables . Then, for any fixed in : These hold wherever the right side expressions make sense (see concept of equality conditional to existence of one side). |
generic point, named functions, point-free notation | Suppose are both functions of variables . Then, for any fixed in : These hold wherever the right side expressions make sense (see concept of equality conditional to existence of one side). |
Statement for directional derivatives
Version type | Statement |
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specific point, named functions | Suppose are both real-valued functions of a vector variable . Suppose is a unit vector. Suppose is a point in the domain of both functions. Then, we have the following product rule for directional derivatives: |
generic point, named functions | Suppose are both real-valued functions of a vector variable . Suppose is a unit vector. Then, we have the following product rule for directional derivatives wherever the right side expression makes sense (see concept of equality conditional to existence of one side): . |
generic point, named functions, point-free notation | Suppose are both real-valued functions of a vector variable . Suppose is a unit vector. Then, we have the following product rule for directional derivatives wherever the right side expression makes sense (see concept of equality conditional to existence of one side): . |
Statement for gradient vectors
Version type | Statement |
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specific point, named functions | Suppose are both real-valued functions of a vector variable . Suppose is a point in the domain of both functions. Then, we have the following product rule for gradient vectors: . Note that the products on the right side are scalar-vector multiplications. |
generic point, named functions | Suppose are both real-valued functions of a vector variable . Then, we have the following product rule for gradient vectors wherever the right side expression makes sense (see concept of equality conditional to existence of one side): . Note that the products on the right side are scalar-vector function multiplications. |
generic point, named functions, point-free notation | Suppose are both real-valued functions of a vector variable . Then, we have the following product rule for gradient vectors wherever the right side expression makes sense (see concept of equality conditional to existence of one side): . Note that the products on the right side are scalar-vector function multiplications. |
Statement for multiple functions
Statement for partial derivatives
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Statement for directional derivatives
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Statement for gradient vectors
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