9.5: The Chain Rule for Multivariable Functions
- Page ID
- 144345
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- Suppose \(\nabla h(3, 0) = (-1, 5)\). Find \(\dfrac{d}{dt} h(2t + 3, t^2 - 4t)\biggr|_{t=0}\).
- Suppose \(\nabla f(0, 0) = (3, -1)\). Find \(\dfrac{d}{ds} f\Bigl(\ln(s^2), s^3 - 4s + 3\Bigr)\biggr|_{s = 1}\).
- Suppose \(\nabla f\left(\sqrt{2}, \dfrac{1}{\sqrt{2}}\right) = (3, -1)\). Find \(\dfrac{d}{d\theta} f\Bigl(\sec \theta \tan \theta, \sin \theta\Bigr)\biggr|_{\theta = \frac{3\pi}{4}}\).
- Suppose \(\nabla g(1, 1) = (3, 4)\). Find \(\dfrac{d}{dt} g\Bigl(3t^2 - 2, \ln t + 1\Bigr)\biggr|_{t = 1}\).
- Suppose \(\nabla h(0, -3, 1) = (1, 2, 3)\). Find \(\dfrac{d}{ds} h(2s^2 + s, s - 3, s^3 + 1)\biggr|_{s = 0}\).
- Suppose \(\nabla g\left(-7, -\dfrac{1}{4}\right) = (1, 0)\). Find \(\dfrac{\partial}{\partial u} g\left(3u - e^{u + v}, \dfrac{1}{u - v}\right)\biggr|_{(u, v) = (-2, 2)}\) and \(\dfrac{\partial}{\partial v} g\left(3u - e^{u + v}, \dfrac{1}{u - v}\right)\biggr|_{(u, v) = (-2, 2)}\).
- Suppose \(\nabla K(-2, -2) = (1, 3)\). Find \(\dfrac{\partial}{\partial u} K\left(uv, \dfrac{u}{v}\right)\biggr|_{(u, v) = (2, -1)}\) and \(\dfrac{\partial}{\partial v} K\left(uv, \dfrac{u}{v}\right)\biggr|_{(u, v) = (2, -1)}\).