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Partial Derivatives
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f(x, y) = e^{x+y}
\frac{\partial f}{\partial x} = e^{x+y}, \frac{\partial f}{\partial y} = e^{x+y}
f(x, y) = x^3 + y^3
\frac{\partial f}{\partial x} = 3x^2, \frac{\partial f}{\partial y} = 3y^2
f(x, y) = e^x \sin(xy)
\frac{\partial f}{\partial x} = e^x \sin(xy) + ye^x \cos(xy), \frac{\partial f}{\partial y} = xe^x \cos(xy)
f(x, y) = x^2y + 3xy^2
\frac{\partial f}{\partial x} = 2xy + 3y^2, \frac{\partial f}{\partial y} = x^2 + 6xy
f(x, y) = \cos^2(x) + \sin^2(y)
\frac{\partial f}{\partial x} = -2\sin(x)\cos(x), \frac{\partial f}{\partial y} = 2\sin(y)\cos(y)
f(x, y) = \cos(x + y)
\frac{\partial f}{\partial x} = -\sin(x + y), \frac{\partial f}{\partial y} = -\sin(x + y)
f(x, y) = \ln(xy)
\frac{\partial f}{\partial x} = \frac{1}{x}, \frac{\partial f}{\partial y} = \frac{1}{y}
f(x, y, z) = xyz
\frac{\partial f}{\partial x} = yz, \frac{\partial f}{\partial y} = xz, \frac{\partial f}{\partial z} = xy
f(x, y) = xye^{x-y}
\frac{\partial f}{\partial x} = ye^{x-y} + xye^{x-y}, \frac{\partial f}{\partial y} = xe^{x-y} - xye^{x-y}
f(x, y) = \tan(xy)
\frac{\partial f}{\partial x} = y(1 + \tan^2(xy)), \frac{\partial f}{\partial y} = x(1 + \tan^2(xy))
f(x, y) = \frac{1}{x^2 + y^2}
\frac{\partial f}{\partial x} = -\frac{2x}{(x^2 + y^2)^2}, \frac{\partial f}{\partial y} = -\frac{2y}{(x^2 + y^2)^2}
f(x, y) = \ln(x^2 + y^2)
\frac{\partial f}{\partial x} = \frac{2x}{x^2 + y^2}, \frac{\partial f}{\partial y} = \frac{2y}{x^2 + y^2}
f(x, y) = \cos(x) \sin(y)
\frac{\partial f}{\partial x} = -\sin(x)\sin(y), \frac{\partial f}{\partial y} = \cos(x)\cos(y)
f(x, y) = \arctan\left(\frac{x}{y}\right)
\frac{\partial f}{\partial x} = \frac{y}{x^2 + y^2}, \frac{\partial f}{\partial y} = -\frac{x}{x^2 + y^2}
f(x, y) = \sqrt{x} + \sqrt{y}
\frac{\partial f}{\partial x} = \frac{1}{2\sqrt{x}}, \frac{\partial f}{\partial y} = \frac{1}{2\sqrt{y}}
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