# JEE Main & Advanced Mathematics Differentiation Definition

## Definition

Category : JEE Main & Advanced

The partial differential coefficient of $f(x,\,y)$ with respect to $x$ is the ordinary differential coefficient of $f(x,\,y)$ when $y$ is regarded as a constant. It is written as $\text{ }\frac{\partial f}{\partial x}$ or ${{D}_{x}}\,f$ or ${{f}_{x}}$.

Thus, $\frac{\partial f}{\partial x}=\underset{h\to 0}{\mathop{\lim }}\,\frac{f(x+h,\,y)-f(x,y)}{h}$

Again, the partial differential coefficient $\frac{\partial f}{\partial y}$ of $f(x,\,y)$ with respect to $y$ is the ordinary differential coefficient of $f(x,\,y)$ when x is regarded as a constant.

Thus, $\frac{\partial f}{\partial y}=\underset{k\to 0}{\mathop{\lim }}\,\frac{f(x,\,y+k)-f(x,\,y)}{k}$

e.g., If $z=f(x,\,y)={{x}^{4}}+{{y}^{4}}+3x{{y}^{2}}+{{x}^{2}}y+x+2y$

Then $\frac{\partial z}{\partial x}$ or $\frac{\partial f}{\partial x}$ or ${{f}_{x}}=4{{x}^{3}}+3{{y}^{2}}+2xy+1$ (Here $y$ is regarded as constant)

$\frac{\partial z}{\partial y}\ \ \text{or}\ \,\frac{\partial f}{\partial y}$ or ${{f}_{y}}=4{{y}^{3}}+6xy+{{x}^{2}}+2$(Here $x$ is regarded as constant)

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