**Category : **JEE Main & Advanced

Let \[f(x,\,y)\] be a function of two variables such that \[\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}\] both exist.

(1) The partial derivative of \[\frac{\partial f}{\partial y}\] w.r.t. \['x'\] is denoted by \[\frac{{{\partial }^{2}}f}{\partial {{x}^{2}}}\text{ }\]or \[{{f}_{xx}}\].

(2) The partial derivative of \[\frac{\partial f}{\partial y}\] w.r.t. \['y'\] is denoted by \[\frac{{{\partial }^{2}}f}{\partial {{y}^{2}}}\] or \[{{f}_{yy}}\].

(3) The partial derivative of \[\frac{\partial f}{\partial x}\] w.r.t. \['y'\] is denoted by \[\frac{{{\partial }^{2}}f}{\partial y\,\partial x}\] or \[{{f}_{xy}}\].

(4) The partial derivative of \[\frac{\partial f}{\partial y}\] w.r.t. \[x\] is denoted by \[\frac{{{\partial }^{2}}f}{\partial y\partial x}\] or \[{{f}_{yx}}\]. These four are second order partial derivatives.

**Note :** If \[f(x,\,y)\] possesses continuous partial derivatives then in all ordinary cases. \[\frac{{{\partial }^{2}}f}{\partial x\,\partial y}=\frac{{{\partial }^{2}}f}{\partial y\,\partial x}\] or \[{{f}_{xy}}={{f}_{yx}}\].

*play_arrow*Introduction*play_arrow*Some Standard Differentiation*play_arrow*Theorems for Differentiation*play_arrow*Methods of Differentiation*play_arrow*Differentiation of a Function with Respect to Another Function*play_arrow*Successive Differentiation or Higher Order Derivatives*play_arrow*\[{{n}^{th}}\] Derivative Using Partial Fractions*play_arrow*Differentiation of Integral Function*play_arrow*Leibnitz?s Theorem*play_arrow*Definition*play_arrow*Higher Partial Derivatives*play_arrow*Euler's Theorem on Homogeneous Functions*play_arrow*Deduction of Euler?s Theorem*play_arrow*Derivative as the Rate of Change

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