### Partial Derivatives Example 3

In multivariable calculus you may be asked to find the partial derivatives. When deriving with respect to a variable, just treat all other variables as a constant.

Find all first partial derivatives of \(f(x,y) = \frac{x}{{{{(x + y)}^2}}}\)

Now you need to use the quotient rule, but when doing this derive with respect to the variable you're finding the partial derivative for! Remember the quotient rule is:

\[{\left[ {\frac{f}{g}} \right]^\prime } = \frac{{gf' - g'f}}{{{g^2}}}\]

Where f is the numerator and g is the denominator of the function you need to find the derivative for. But when finding f' and g', you do it with respect to the variable you're finding the partial derivative for.

Let's find the partial derivative with respect to x first, and we'll show all examples since the quotient rule involves a bit of arithmetic. Try to factor when possible to reduce the power of the denominator.

\[\frac{{\partial f}}{{\partial x}} = \frac{{{{(x + y)}^2}(1) - 2(x + y)x}}{{{{(x + y)}^4}}}\]

\[\frac{{\partial f}}{{\partial x}} = \frac{{(x + y - 2x)(x + y)}}{{{{(x + y)}^4}}}\]

\[\frac{{\partial f}}{{\partial x}} = \frac{{y - x}}{{{{(x + y)}^3}}}\]

There we go, it's a bit of arithmetic but we should always try to simplify as much as reasonably possible. Now let's find the partial derivative with respect to y, and the quotient rule is the same except our derivatives are found with respect to the "y" variable instead.

\[\frac{{\partial f}}{{\partial y}} = \frac{{{{(x + y)}^2}(0) - 2(x + y)x}}{{{{(x + y)}^4}}}\]

\[\frac{{\partial f}}{{\partial y}} = \frac{{ - 2x(x + y)}}{{{{(x + y)}^4}}}\]

\[\frac{{\partial f}}{{\partial y}} = \frac{{ - 2x}}{{{{(x + y)}^3}}}\]

Please see our other examples on partial derivatives, including chain rule, product rule, quotient rule, and multivariable implicit differentiation:

Partial Derivatives Examples:

Partial Derivatives (Ex1)

Partial Derivatives (Ex2)

Partial Derivatives (Ex4)

Multivariable Chain Rule Examples:

Multivariable Chain Rule (Ex1)

Multivariable Chain Rule (Ex2)

Implicit Multivariable Differentiation:

Multivariable Implicit Differentiation

Return to the multivariable differentiation hub page:

Partial Derivatives, Multivariable Differentiation