What is the derivative of the function f = x^2 + xy + y^2 with respect to y?

To find the derivative of the function ( f = x^2 + xy + y^2 ) with respect to ( y ), we need to apply the rules of differentiation carefully, considering each term in the function.

The first term, ( x^2 ), is a constant with respect to ( y ), so its derivative is zero.

Next, when differentiating ( xy ) with respect to ( y ), we treat ( x ) as a constant (since the differentiation is with respect to ( y )). The derivative of ( xy ) is ( x ).

Finally, for the term ( y^2 ), we differentiate it normally with respect to ( y), yielding ( 2y ).

Adding these derivatives together, we get:

• From ( x^2 ): ( 0 )

• From ( xy ): ( x )

• From ( y^2 ): ( 2y )

Thus, the total derivative with respect to ( y ) is ( x + 2y ).

This result corresponds to the correct answer, confirming that the derivative of the function with respect to ( y ) is indeed ( x + 2y ).