The partial derivative of xy with respect to x is equal to y and the partial derivative of xy with respect to y is equal to x. Their formula is given below:

- $\dfrac{\partial }{\partial x}(xy)=y$
- $\dfrac{\partial }{\partial y}(xy)=x$

## Partial Derivative of xy with respect to x

Let us will find the partial derivative of xy using the definition.

Let f(x, y) = xy.

By definition of partial derivatives, we have

$\dfrac{\partial }{\partial x}(xy)$ = lim_{h→0} $\dfrac{f(x+h, y)- f(x,y)}{h}$

= lim_{h→0} $\dfrac{(x+h)y- xy}{h}$

= lim_{h→0} $\dfrac{xy+hy- xy}{h}$

= lim_{h→0} $\dfrac{hy}{h}$

= lim_{h→0} y

= y

So the partial derivative of xy with respect to x is y and this is obtained by the definition of partial derivatives.

**Also Read:** Derivative of x/y

**Question 1:** If z=xy, then find ∂z/∂x.

**Answer:**

To find ∂z/∂x, that is, the partial derivative of xy with respect to x, we will treat y as a constant and x as a variable. That is,

∂z/∂x = $\dfrac{\partial }{\partial x}(xy)$ = $y \dfrac{d}{dx}(x)$ = $y$.

Thus, if z=xy then ∂z/∂x=y.

## Partial Derivative of xy with respect to y

By the definition of partial derivatives, we have

$\dfrac{\partial }{\partial y}(xy)$ = lim_{k→0} $\dfrac{f(x, y+k)- f(x,y)}{k}$ where f(x, y) = xy.

= lim_{k→0} $\dfrac{x(y+k)- xy}{k}$

= lim_{k→0} $\dfrac{xy+xk- xy}{k}$

= lim_{k→0} $\dfrac{xk}{k}$

= lim_{k→0} x

= x

So the partial derivative of xy with respect to y is x and this is obtained by the definition of partial derivatives.

**Question 2:** If z=xy, then find ∂z/∂y.

**Answer:**

While finding ∂z/∂y, that is, the partial derivative of xy with respect to y, we will treat x as a constant and y as a variable. So we have

∂z/∂y = $\dfrac{\partial }{\partial y}(xy)$ = $x \dfrac{d}{dy}(y)$ = $x$.

Therefore, if z=xy then ∂z/∂y = x.

## FAQs

**Q1: What is the partial derivative of xy with respect to x?**

Answer: The partial derivative of xy with respect to x is equal to y.

**Q2: What is the partial derivative of xy with respect to y?**

Answer: The partial derivative of xy with respect to y is equal to x.