Partial and Total Differentiation of Multivariate Functions
A multivariate function may be differentiated with respect to each variable, which is called partial differentiation. By combining all the partial differentiations, we define total differentiation. The essence of (total) differentiation is a linear approximation. In the case of a univariate function, we approximate the function y = f(x) in the neighbor of a point, say x=a, by the tangent line y = f'(a)(x-a) + f(a). In the case of a multivariate function, we approximate the function y = f(x1, x2,...,xn) in the neighbor of a point, say a=(a1, a2, ..., an), by the tangent hyperplane at the point x=a.
Lecture notes: https://bruneimathclub.blogspot.com/2...
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