1. **State the problem:** Find the partial derivative of the function $f(x,y) = x^2 y + \cos(x)$ with respect to $x$. 2. **Recall the formula:** The partial derivative of $f(x,y)$ with respect to $x$ is denoted as $f_x = \frac{\partial}{\partial x} f(x,y)$, treating $y$ as a constant. 3. **Apply the derivative rules:** - The derivative of $x^2 y$ with respect to $x$ is $2x y$ since $y$ is constant. - The derivative of $\cos(x)$ with respect to $x$ is $-\sin(x)$. 4. **Combine the results:** $$f_x = 2xy - \sin(x)$$ 5. **Final answer:** The partial derivative of $f(x,y)$ with respect to $x$ is $$f_x = 2xy - \sin(x)$$