9.3: Partial Derivatives
- Page ID
- 144343
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- Using the graph of \(f(x, y)\) is above. Determine if each of the following partial derivatives is positive, negative, or zero:
- \(f_x(1, 1)\)
- \(f_y(1, 1)\)
- \(f_x(1, -1)\)
- \(f_y(1, -1)\)
- \(f_x(0, 0)\)
- \(f_x(1, 1)\)
- Given \(z = x + y\), find \(\dfrac{\partial z}{\partial x}\) and \(\dfrac{\partial z}{\partial y}\).
- Given \(z = xy\), find \(\nabla z\).
- Given \(f(x, y) = \sin(3x)\cos(3y)\), find \(f_x(x, y)\) and \(f_y(x, y)\).
- Given \(g(x, y) = \ln(x^6 + y^4)\), find \(\nabla g(x, y)\).
- Given \(h(x, y) = \tan\left(\dfrac{x}{y}\right)\), find \(\dfrac{\partial h}{\partial x}\) and \(\dfrac{\partial h}{\partial y}\).
- Given \(z = \cos(x^2 y) + y^3\), find \(\dfrac{\partial z}{\partial x}\) and \(\dfrac{\partial z}{\partial y}\).
- Given \(z = \sqrt{x^2 + y^2}\), find \(z_x\) and \(z_y\).
- Given \(f(x, y) = \dfrac{xy}{x^2 + y}\), find \(f_x\) and \(f_y\).
- Given \(z = x^2 + 3xy + 2y^2\), find \(\dfrac{\partial^2 z}{\partial x^2}\), \(\dfrac{\partial^2 z}{\partial y^2}\), \(\dfrac{\partial^2 z}{\partial x\partial y}\), and \(\dfrac{\partial^2 z}{\partial y \partial x}\).
- Given \(g(x, y) = e^{x^2 + y^2}\), find \(g_{xx}\), \(g_{yy}\), \(g_{xy}\), and \(g_{yx}\).
- Given \(w(x, y, z) = e^{-2x}\sin(z^2 y)\), find \(\dfrac{\partial w}{\partial x}\), \(\dfrac{\partial w}{\partial y}\), and \(\dfrac{\partial w}{\partial z}\).
- Given \(\zeta(x, y, z) = xy^2 + yz^2 + xz^2\), find \(\nabla \zeta(x, y, z)\).
- Given \(F(x, y, z) = x^3yz^2 - 2x^2yz + 3xz - 2y^3z\), find \(F_{xyz}\), \(F_{zxy}\), and \(F_{yzx}\).