Derivative of z=siny+y^2-x2y

1. Differentiate $- x_{2} y + y^{2} + \sin{\left(y \right)}$ term by term:

   1. The derivative of sine is cosine:

      $$\frac{d}{d y} \sin{\left(y \right)} = \cos{\left(y \right)}$$

   2. Apply the power rule: $y^{2}$ goes to $2 y$

   3. The derivative of a constant times a function is the constant times the derivative of the function.

      1. The derivative of a constant times a function is the constant times the derivative of the function.

         1. Apply the power rule: $y$ goes to $1$

         So, the result is: $x_{2}$

      So, the result is: $- x_{2}$

   The result is: $- x_{2} + 2 y + \cos{\left(y \right)}$

The answer is:

$$- x_{2} + 2 y + \cos{\left(y \right)}$$