JEE Main & Advanced Mathematics Differentiation Definition

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)