1. **State the problem:** Find the partial derivative of the function $F(x,y) = x \tan y + y^2 \cos x$ with respect to $x$, denoted as $F_x$. 2. **Recall the formula:** The partial derivative with respect to $x$ means we treat $y$ as a constant and differentiate only with respect to $x$. 3. **Differentiate each term:** - For $x \tan y$, since $\tan y$ is constant with respect to $x$, the derivative is $\tan y$. - For $y^2 \cos x$, $y^2$ is constant, and the derivative of $\cos x$ with respect to $x$ is $-\sin x$, so the derivative is $y^2 (-\sin x) = -y^2 \sin x$. 4. **Combine results:** $$F_x = \tan y - y^2 \sin x$$ 5. **Final answer:** The partial derivative of $F$ with respect to $x$ is $$F_x = \tan y - y^2 \sin x$$