How to show $ \|fg\|_{L^2}\leq \|\partial_{x_1}f\|_{L^2}\|g\|_{L^2}+\|f\|_{L^2}\|\partial_{x_2}g\|_{L^2} $?

Let $ f,g\in C_c^1(\mathbb{R}^2) $, show that$$\|fg\|_{L^2}\leq \|\partial_{x_1}f\|_{L^2}\|g\|_{L^2}+\|f\|_{L^2}\|\partial_{x_2}g\|_{L^2}.$$

Here is my try. Without loss of generality, we can assume that $ \operatorname{supp}(f,g)\subset\mathbb{R}_+\times\mathbb{R}_+ $. Then I want to use the fundamental theorem of Calculus to represent $ fg $ by the $ \partial_{x_1}fg $ and $ f\partial_{x_2}g $. However I have some difficulty in this procedure. Can you give me some references or hints?