JEE Main & Advanced Mathematics Differentiation Higher Partial Derivatives

 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 \[{{f}_{xy}}={{f}_{yx}}\].