In Schaum Outline Series (Lagrangian dynamics) page 41 it states, with $\normalsize x = x(q\_1,q\_2)$ with $\normalsize q\_1, q\_2$ being generalised coordinates and $\dot{x}$, $\dot{q\_1}$, $\dot{q\_2}$ being the first derivative of $\normalsize x$, $q\_1$, $q\_2$ relative to t:

$$\dot{x} = \frac {\partial x}{\partial q\_1} \dot{q\_1} + \frac {\partial x}{\partial q\_2} \dot{q\_2}\tag{1}$$

The book then says differentiating this partially with respect to $\normalsize\dot{q\_1}$ (1) becomes :

$$\frac {\partial \dot{x}}{\partial \dot{q\_1}} = \frac {\partial x}{\partial q\_1}\tag{2} $$

However I can't see how (2) is derived from the full partial derivative chain rule applied to (1) with respect to $\normalsize\dot{q\_1}$:

$$\frac {\partial \dot{x}}{\partial \dot{q\_1}} = \dot{q\_1} \frac {\partial }{\partial \dot{q\_1}}( \frac {\partial x}{\partial q\_1})+ \frac {\partial \dot{q\_1}}{\partial \dot{q\_1}} \frac {\partial x}{\partial q\_1} + \dot{q\_2} \frac {\partial }{\partial \dot{q\_1}}( \frac {\partial x}{\partial q\_2})+ \frac {\partial \dot{q\_2}}{\partial \dot{q\_1}} \frac {\partial x}{\partial q\_2} \tag{3} $$

giving more terms on the right hand side than in (2)

$$\frac {\partial \dot{x}}{\partial \dot{q\_1}} = \dot{q\_1} \frac {\partial }{\partial \dot{q\_1}}( \frac {\partial x}{\partial q\_1})+ \frac {\partial x}{\partial q\_1} + \dot{q\_2} \frac {\partial }{\partial \dot{q\_1}}( \frac {\partial x}{\partial q\_2})+ \frac {\partial \dot{q\_2}}{\partial \dot{q\_1}} \frac {\partial x}{\partial q\_2} \tag{4} $$