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11 Partial derivatives and multivariable chain rule

11 Partial derivatives and multivariable chain Basic defintions and the Increment TheoremOne thing I would like to point out is that you ve been taking Partial derivativesall your calculus-life. When you computedf/dtforf(t)=Ce kt, you get Cke ktbecauseCandkare constants. The notationdf/dttells you thattis the variablesand everything else you see is a constant. If we use the notationf0instead, then weare relying on your knowing which is the independent variable. It s usually calledsomething like t , not C or k , but every now and then we end up computingdf/dkordf/dC, so watch out! The only rule is: everyone should understand whichis the independent now, studying Partial derivatives , the only di erence is that the other variablesaren t constants they vary but you treat them as constants anyway. It s not a bigdi erence because really, what is a constant? It s always possible to imagine somequantity changing.

the chain rule gives df dx = @f @x + @f @y ·y0. (11.3) The notation really makes a di↵erence here. Both df /dx and @f/@x appear in the equation and they are not the same thing! Derivative along an explicitly parametrized curve One common application of the multivariate chain rule is when a point varies along

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