# JEE Main & Advanced Mathematics Differentiation Higher Partial Derivatives

## Higher Partial Derivatives

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 ${{f}_{xy}}={{f}_{yx}}$.

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