 Homework Statement

Suppose that ##\alpha,\beta: \mathbb{R}\to \mathbb{R}## and ##g: \mathbb{R}^2\to \mathbb{R}## are differentiable functions and ##f: \mathbb{R}\to \mathbb{R}^2## is the function defined by ##f(t) = (\alpha(t),\beta(t))##. Suppose further that
$$f(6) = (10,10), \quad \alpha'(6) = 4, \quad \beta'(6) = 3, \quad g_x(10,10) = 1, \quad \text{and} \quad g_y(10,10) = 2.$$
Then ##(g\circ f)'(6)## equals to
1. ##24##.
2. ##24##.
3. ##2##.
4. ##0##.
5. ##2##.
 Homework Equations
 Multivariate chain rule formula.
The solution is 3: It's just ##(g\circ f)'(6) = (1,2)\cdot (4,3) = (1\times 4)+((2)\times (3)) = 4+6 = 2## using the multivariate chain rule and the dot product.
Is this correct and if not how do I go about doing it?
Thanks!
Is this correct and if not how do I go about doing it?
Thanks!