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. \[ \begin{array}{l} z(x+h, y)=x^{\wedge} 2+2 x h+h^{\wedge} 2-9 x y-9 h y+y^{\wedge} \\ \text { Correct response: } x^{\wedge} 2+2^{*} x^{*} h+h^{\wedge} 2-9^{*} x^{*} y-9^{*} h^{*} y+y^{\wedge} 2 \end{array} \] Using Equation 4.12 from the text, calculate, \[ \frac{\partial z}{\partial x}=\lim _{h \rightarrow 0} \frac{z(x+h, y)-z(x, y)}{h} \] \[ \frac{\partial z}{\partial x}= \] \( \square \)

To find \( z(x+h, y) \), substitute \( x+h \) into the expression for \( z \): \[ z(x+h, y) = (x+h)^2 - 9(x+h)y + y^2 \] Expanding this gives: \[ = x^2 + 2xh + h^2 - 9xy - 9hy + y^2 \] Combining like terms results in: \[ = x^2 - 9xy + y^2 + 2xh - 9hy + h^2 \] Next, we can proceed to calculate the partial derivative using the limit definition. Now, we have: \[ z(x, y) = x^2 - 9xy + y^2 \] Thus, \[ z(x+h, y) - z(x, y) = (x^2 - 9xy + y^2 + 2xh - 9hy + h^2) - (x^2 - 9xy + y^2) \] This simplifies to: \[ = 2xh - 9hy + h^2 \] Now, we can substitute this back into the limit definition: \[ \frac{\partial z}{\partial x} = \lim_{h \rightarrow 0} \frac{2xh - 9hy + h^2}{h} \] Dividing everything by \( h \): \[ = \lim_{h \rightarrow 0} (2x - 9y + h) \] Taking the limit as \( h \) approaches 0 gives: \[ \frac{\partial z}{\partial x} = 2x - 9y \] Thus, the answer is: \[ \frac{\partial z}{\partial x} = 2x - 9y \, \square \]