Calculate the partial derivative using the limit definitions only. In this adaptive problem, we walk through the steps of solving the problem. \[ \frac{\partial z}{\partial x} \text { for } z=x^{2}-9 x y+y^{2} \] Find \( z(x+h, y) \). Expand and simplify the expression. \( z(x+h, y)=\square \) Section Attempt 1 of 999 Verify

To find \( z(x+h, y) \), we substitute \( x+h \) into the equation \( z \) which is given as \( z = x^2 - 9xy + y^2 \). So, we have: \[ z(x+h, y) = (x+h)^2 - 9(x+h)y + y^2 \] Now let's expand that: 1. Expanding \( (x+h)^2 \) gives us \( x^2 + 2xh + h^2 \). 2. Expanding \( -9(x+h)y \) gives us \( -9xy - 9hy \). Putting it all together: \[ z(x+h, y) = (x^2 + 2xh + h^2) - 9xy - 9hy + y^2 \] \[ = x^2 - 9xy + y^2 + 2xh + h^2 - 9hy \] Now combine like terms: \[ z(x+h, y) = z(x,y) + 2xh + h^2 - 9hy \] So your final expression for \( z(x+h, y) \) is: \[ z(x+h, y) = z(x, y) + 2xh + h^2 - 9hy \] Now, you can simply plug this back into the limit definition to find \( \frac{\partial z}{\partial x} \). Have fun calculating!