{
  "problem": "$f(v, w) = (v, v^2, w^3)$ What is $\\dfrac{\\partial f}{\\partial v}$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(1, 2v, 0)$ (Choice B) B $(0, 0, 3w^2)$ (Choice C) C $(0, 0, 0)$ (Choice D) D $(1, 2v, 3w^2)$",
  "hints": [
    "The partial derivative of a vector valued function is component-wise partial differentiation. $\\begin{aligned} &f(v, w) = (f_0(v, w), f_1(v, w), f_2(v, w)) \\\\ \\\\ &f_v = \\left( \\dfrac{\\partial f_0}{\\partial v}, \\dfrac{\\partial f_1}{\\partial v}, \\dfrac{\\partial f_2}{\\partial v} \\right) \\\\ \\\\ &f_w = \\left( \\dfrac{\\partial f_0}{\\partial w}, \\dfrac{\\partial f_1}{\\partial w}, \\dfrac{\\partial f_2}{\\partial w} \\right) \\end{aligned}$",
    "Because we're taking a partial derivative with respect to $v$, we'll treat $w$ as if it were a constant.",
    "Therefore, $f_v = (1, 2v, 0)$."
  ]
}
{'source': 'AMPS', 'file': 'amps/khan/partial-derivatives-of-vector-valued-functions/16077396204.json'}