Let us assume a function

*f*with two variables, i.e., f(x, y). Keeping y as constant and differentiating f with respect to x, we get the partial derivative of*f*function with respect to x, denoted as $\frac{\partial f}{\partial x}$ or $f_x$.Similarly, keeping x as constant and differentiating f with respect to y, we get the partial derivative of

*f*function with respect to y, denoted as $\frac{\partial f}{\partial y}$ or $f_y$.Plug in the function with two variables into the partial derivative calculator to get the partial derivative of the given function.

**Step 1 :**Note down the given

*f*function with their two variables x and y.

**Step 2 :**Find the partial derivative according to the problem, if it is with respect to x, then find $\frac{\partial f}{\partial x}$ or if it is with respect to y, then find $\frac{\partial f}{\partial y}$.

### Solved Examples

**Question 1:**If $f(x,y)$ = $x^{2}y + 2x$, find the partial derivative $f_x$.

**Solution:**

**Step 1 :**Given : $f(x,y)$ = $x^{2}y+2x$.

**Step 2 :**$\frac{\partial f}{\partial x}$ = $\frac{\partial }{\partial x}$$(x^{2}y+2x)$

= $\frac{\partial }{\partial x}$$(x^{2}y)$ + $\frac{\partial }{\partial x}$$(2x)$

= $(2xy) + (2)$ = $2xy + 2$

Therefore, partial derivative of $x^{2}y+2x$ = $2xy + 2$.

**Question 2:**if $f(x,y)$ = $xe^{xy}$, find the partial derivative $f_y$.

**Solution:**

**Step 1 :**Given : $f(x,y)$ = $xe^{xy}$

**Step 2 :**$\frac{\partial f}{\partial y}$ = $\frac{\partial }{\partial y}$$(xe^{xy})$

= $(x)(xe^{xy})$ = $x^{2}e^{xy}$

Therefore, partial derivative of $xe^{xy}$ = $x^{2}e^{xy}$.