Hey guys!
I am supposed to show that the function f(x,y) = x^2 + y^3 is differentiable and to determine the derivate.
Do i have to show this with the differential quotient? I know about partial derivates but how to determine the "entire derivate" f'(x,y)?
I know lots of questions and no solution. Sorry for that, but please give me a clue, some keywords at least.
Thanks!
Inf
That's simply not true- but of course, you have to generalize the notion of "derivative". If f is a function from to then the derivative of f, at point p, is the linear function from to , that best approximates f in a neighborhood of p. Specifically, we say that such a function is differentiable at p if and only if there exist a linear function L and a function , from to , such that
Notice the the derivative is defined as a linear transformation. In the case of to , it would be the function where is the "usual" derivative at p. In the case of to the linear transformation is maps to . In other words it is the dot product of and p. That's why most texts treat as being "the" derivative.