Q: Suppose that the functions $f : \mathbb{R^2} \to \mathbb{R}$, $g : \mathbb{R^2} \to \mathbb{R}$ and $h : \mathbb{R^2} \to \mathbb{R}$ are
continuously differentiable. Express the following two limits in terms of partial derivatives of these functions:
a. $\lim_{t \to 0} \dfrac{f(g(1 + t, 2), h(1 + t, 2)) - f(g(1, 2), h(1, 2)) }{t}$
b. $\lim_{t \to 0} \dfrac{f(g(1, 2) + t, h(1, 2)) - f(g(1, 2), h(1, 2))}{t}$
A: a. $\dfrac{\partial f}{\partial g} (g(1,2)) \dfrac{\partial g}{\partial x} ((1,2)) + \dfrac{\partial f}{\partial h} (h(1,2)) \dfrac{\partial h}{\partial x} ((1,2)) \ ??$
b. $\dfrac{\partial f}{\partial g} (g(1,2)) \ ??$
Did I get the answers right?
a)
$$\dfrac{\partial f}{\partial g} (g(1,2),h(1,2)) \dfrac{\partial g}{\partial x} ((1,2)) + \dfrac{\partial f}{\partial h} (g(1,2),h(1,2)) \dfrac{\partial h}{\partial x} ((1,2)) \ $$
b)
$$\dfrac{\partial f}{\partial x} (g(1,2),h(1,2)) $$