The formula ∂f/∂x = lim(h → 0) [f(x + h, y) - f(x, y)]/h represents the partial derivative of f with respect to x, and it's evaluated at the point (x, y).
Cauchy's Mean Value Theorem states that if two functions f(x) and g(x) are continuous on the closed interval [a, b] and differentiable on the open interval (a, b), and g'(x) ≠0 for any x in (a, b), then there exists a point c in (a, b) such that:
(f'(c) / g'(c)) = (f(b) - f(a)) / (g(b) - g(a))
The necessary conditions for Cauchy's Mean Value Theorem are:
1. The functions f(x) and g(x) are continuous on [a, b].
2. The functions f(x) and g(x) are differentiable on (a, b).
