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Unformatted text preview: Lecture 6 : Friday April 10th firstname.lastname@example.org Key concepts : Vector valued function, gradient, differentiability, properties, chain rule 6.1 Basic Properties of derivatives. For functions of one variable f : R R , the basic properties of derivatives stem from the basic properties of limits, namely the sum, product and quotient rules, as well as the chain rule. The difference for functions of more than one variable is that for these properties to hold, the function is required to be differentiable . If the function is not differentiable then the rules can fail. Here are the basic properties of derivatives. For all of the following rules, f,g : R n R m are defined on an open set U and differentiable functions at a point a U . For the quotient rule, g is non-zero on U . Constant multiple rule: cf ( x ) is differentiable at a and ( cf )( a ) = c f ( a ) . Sum rule: f ( x ) + g ( x ) is differentiable at a and ( f + g )( a ) = f ( a ) + g ( a ) ....
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