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Unformatted text preview: Lecture 5 : Wednesday April 8th email@example.com Key concepts : Vector valued function, gradient, differentiability, properties, chain rule 5.1 Derivatives of vector-valued functions We have not until now discussed derivatives of functions f : R n R m . In order to state the rules, we need some notation. A function f : R n R m assigns to each vector x = ( x 1 ,x 2 ,...,x n ) a new vector ( f 1 ( x ) ,f 2 ( x ) ,...,f m ( x )). The derivative of f if it exists can be defined in terms of limits. However, for our purposes, we only need to know that f j ( x ) = ( f 1 j ( x ) ,f 2 j ( x ) ,...,f mj ( x )) . Remember f j ( x ) means the derivative of f with respect to x j . So f 1 j ( x ) is the derivative of f 1 ( x ) with respect to x j . Thus the derivative is found by taking derivatives of each of the components of f ( x ). Example. Let f ( x,y ) = ( x + y,x/y,x- y ). This is a function f : R 2 R 3 . Then f 1 ( x,y ) = (1 , 1 /y, 1) and f 2 ( x,y ) = (1 ,- x/y 2 ,- 1) ....
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