2. a) Define what it means to describe a function f of two real variables as differentiable at (a, b)? Define (as limits) the partial derivatives df/dx and df/dy at
(a, b) and prove that if f is differentiable at (a, b) then both these partial derivatives exist.
it this correct for 2. a)
f is differentiable at (a,b) iff there exists a linear mapping L such that lim((f(a+h,b+k)-f(a,b)-L(h,k))/|(h,k)|,(h,k)->(0,0))=0. ∂f/∂x(a,b)=lim((f(a+h,b)-f(a,b))/h,h->0), and ∂f/∂y(a,b)=lim((f(a,b+k)-f(a,b))/k,k->0).
does L=grad f; that is, L(h,k)=(∂f/∂x,∂f/∂y).(h,k)=h*∂f/∂x+k*∂f/∂y.
what do i do next?
b) Prove from the definition in a) that the function f defined by f(x,y) =xy(x+y) is differentiable at every point of itís domain
c) If g(x, y) = xy prove that g is not differentiable at (0, b) for any non- zero value of b
d) Prove that the function g of part c) is differentiable at all points (a,b) for which a is not zero and at the origin (0,0)
Am mainly stuck on parts c) and d) but am unsure if am an starting along the right vein with the first parts