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Verify $\large\frac{\partial ^{2} y}{\partial x\partial y}=\frac{\partial ^{2} y}{\partial y\partial x}$ for the following function;$\;u=\sin 3x\cos4y$

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  • If $u=f(x,y)$ is the function of two independent variables then
  • $\large\frac{\partial u}{\partial y (x_0,y_0)}=\frac{d}{dy}$$f(x_0,y)\;and \;\large\frac{\partial u}{\partial x(x_0,y_0)}=\large\frac{d}{dy}$$ f(x,y_0)$ Provided they exist
  • The Second order parral derivaties are $\large\frac{\partial}{\partial x} \frac{\partial u}{\partial y}=\frac{\partial ^2 u}{\partial x \partial y},\frac{\partial}{\partial y} \frac{\partial u}{\partial y}=\frac{\partial ^2 u}{\partial y^2},\frac{\partial}{\partial y} \frac{\partial u}{\partial x}=\frac{\partial ^2 u}{\partial y \partial x},$$\;and\;\large\frac{\partial}{\partial x} \frac{\partial u}{\partial x}=\frac{\partial ^2 u}{\partial x ^2}$ Partial derivatives of functions of more similarly defined
Given $u=\sin 3x \cos 4y$
Step 1:
$\large\frac{\partial u}{\partial y}$$=-4 \sin 3x \sin 4y$
$\large\frac{\partial^2 u}{\partial x \partial y}$$=-12 \cos 3x \sin 4 y$
Step 2:
$\large\frac{\partial u}{\partial x}$$=3 \cos 3x \cos 4y$
$\large\frac{\partial^2 u}{\partial y \partial x}$$=-12 \cos 3x \sin 4 y$
Step 3:
$\large\frac{\partial^2 u}{\partial x \partial y}=\frac{\partial^2 u}{\partial y \partial x}$
answered Aug 12, 2013 by meena.p

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