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    Using the Chain rule for derivatives in higher dimension

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    Let f(u,v,w) = (eu-w, cos(v+u)+sin(u+v+w)) and g(x,y) = (ex, cos(y-x), e-y). Calculate f o g and D(f o g)(0,0).

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    Given a function from R^n to R^m, the derivative is no longer simply a function as it is in introductory calculus. Instead, the derivative at a point is a linear map, which can be represented by a matrix of partial derivatives. There is a higher dimensional analogue of the 1 dimensional Chain rule which can be used to calculate the derivative map of a composite function. The solution comprises a one page Word document with equations written in Mathtype illustrating the use of the higher dimensional Chain rule.