# JEE Main & Advanced Mathematics Differentiation Deduction of Euler?s Theorem

## Deduction of Euler?s Theorem

Category : JEE Main & Advanced

If $f(x,\,y)$ is a homogeneous function in $x,\,\,y$ of degree $n,$ then

(1) $x\frac{{{\partial }^{2}}f}{\partial {{x}^{2}}}+y\frac{{{\partial }^{2}}f}{\partial x\,\partial y}=(n-1)\,\frac{\partial f}{\partial x}$

(2) $x\frac{{{\partial }^{2}}f}{\partial y\,\partial x}+y\frac{{{\partial }^{2}}f}{\partial {{y}^{2}}}=(n-1)\,\frac{\partial f}{\partial y}$

(3) ${{x}^{2}}\frac{{{\partial }^{2}}f}{\partial {{x}^{2}}}+2xy\frac{{{\partial }^{2}}f}{\partial x\,\partial y}+{{y}^{2}}\frac{{{\partial }^{2}}f}{\partial {{y}^{2}}}=n(n-1)\,f(x,\,y)$

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