![]() ![]() Note that the products on the right side are scalar-vector function multiplications. 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): Suppose are both real-valued functions of a vector variable. Note that the products on the right side are scalar-vector multiplications. Then, we have the following product rule for gradient vectors: Suppose is a point in the domain of both functions. Statement for gradient vectors Version type 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): Suppose are both real-valued functions of a vector variable. The Product Rule says that the derivative of a product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function. Then, we have the following product rule for directional derivatives: Now, lets differentiate the same equation using the chain rule which states that the derivative of a composite function equals: (derivative of outside) (inside) (derivative of inside). The derivative of f(x) c where c is a constant is given by f (x) 0 Example f(x) - 10, then f (x) 0 2 - Derivative of a power function (power. Before using the chain rule, lets multiply this out and then take the derivative. The basic rules of Differentiation of functions in calculus are presented along with several examples. Suppose are both real-valued functions of a vector variable. Rules of Differentiation of Functions in Calculus. Statement for directional derivatives Version type Suppose the partial derivatives and both exist. Suppose is a point in the domain of both and. ![]() Statement for partial derivatives for functions of multiple variables Version type Generic point, named functions, point-free notation These hold wherever the right side expressions make sense (see concept of equality conditional to existence of one side). The derivatives used here are partial derivatives. Statement for two functions Statement for partial derivatives for functions of two variables
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