Is there any formal calculation method for partial derivative?

Question

I am learning undergraduate level PDE. The massive calculation of partial derivatives often confuse me a lot. Though I can understand them after some careful check, I am curious about how mathematician find these 'strange' derivative deformation? Is there any formal calculation method for doing these partial derivatives? For example$$\frac{\partial u}{\partial t}\cdot\frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial}{\partial x}(\frac{\partial u}{\partial t}\cdot\frac{\partial u}{\partial x})-\frac{\partial }{\partial x}(\frac{\partial u}{\partial t})\cdot\frac{\partial u}{\partial x}\\=\frac{\partial }{\partial x}(\frac{\partial u}{\partial t}\cdot\frac{\partial u}{\partial x})-\frac{1}{2}\cdot\frac{\partial}{\partial t}(\frac{\partial u}{\partial x})^{2}$$