# Confusion with metric components in spherical coordinate system

I know that by definition the transition between basis vectors $e_{a}$ of one coordinate system to basis vectors $e_{a^\prime}$ of another coordinate systems is performed via the next expression: $$e_{a^\prime}=\Lambda^b_{a^\prime}e_b$$ $$\Lambda^b_{a^\prime}=\frac{\partial x^{b}}{\partial x^{a^\prime}}$$ $$e_b=e_a$$ But when I want to find basis vectors of spherical coordinate system via cartesian coordinate system by the definition above I come across the next problem:

The spherical system coordinates expressed as $x=rsin{\theta}cos{\phi}, y=rsin{\theta}sin{\phi}, z=rcos{\theta}$, so I am performing the steps which are shown in definition:$$\partial r=\frac{\partial x}{\partial r}\partial x +\frac{\partial y}{\partial r}\partial y + \frac{\partial z}{\partial r}\partial z$$ But according to my book (Relativity demystified) this is not correct and should be done in vice-versa way namely: $$\partial r=\frac{\partial r}{\partial x}\partial x +\frac{\partial r}{\partial y}\partial y + \frac{\partial r}{\partial z}\partial z$$ But I see no sense in their action and how they got to the correct answer. Is it misprint or my misunderstanding?

This is really a mistake. Relativity Demystified, example 2-5.