1. The problem is to find the partial derivative of the function $f(x,y) = -4x + 2y$ with respect to $x$. 2. The formula for the partial derivative of a function $f(x,y)$ with respect to $x$ is: $$\frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x+h,y) - f(x,y)}{h}$$ 3. Important rule: When taking the partial derivative with respect to $x$, treat $y$ as a constant. 4. Apply the derivative to each term: - The derivative of $-4x$ with respect to $x$ is $-4$. - The derivative of $2y$ with respect to $x$ is $0$ since $y$ is treated as a constant. 5. Therefore, the partial derivative is: $$\frac{\partial f}{\partial x} = -4 + 0 = -4$$ Final answer: $$\boxed{-4}$$