How to get this Production Function in Growth Rates
Given
$$Y = Af(K,L,Z)$$
it follows that
$$\dot Y = \dot A f + A\frac{\partial f}{\partial K}\dot K + A \frac{\partial f}{\partial L} \dot L + A \frac{\partial f}{\partial Z} \dot Z,$$
where dotted expression are time derivatives and dividing with $Y$ it follows
$$\frac{\dot Y}{Y} = \frac{\dot A}{A}  + \left[A\frac{\partial f}{\partial K}\right]\frac{\dot K}{Y}+ \left[A \frac{\partial f}{\partial L} \frac{1}{Y}\right] \dot L + \left[A \frac{\partial f}{\partial Z} \frac{1}{Y}\right]\dot Z ,$$
and dividing and multiplying with $L$ and $Z$ in relevant summands it follows that
$$\frac{\dot Y}{Y} = \frac{\dot A}{A}  + \left[A\frac{\partial f}{\partial K}\right]\frac{\dot K}{Y}+ \left[A \frac{\partial f}{\partial L} \frac{L}{Y}\right] \frac{\dot L}{L} + \left[A \frac{\partial f}{\partial Z} \frac{Z}{Y}\right]\frac{\dot Z}{Z} ,$$
and then terms in square brackets are defined as $\alpha_1,\alpha_2$ and $\alpha_3$ being respectively

The marginal productivity of capital $\alpha_1 = A \frac{\partial f}{\partial K}$
The elasticity of output with respect to labor $\alpha_2 = A \frac{\partial f}{\partial L} \frac{L}{Y}$ and
The elasticity of output with respect to other factors  $\alpha_3 = A \frac{\partial f}{\partial Z} \frac{Z}{Y}$

In the text the authors write $\partial Y/\partial X$ with $X\in \{K,L,Z\}$ instead of
$\partial f/\partial X$ which must be a typo. For example it is hard to convince yourself that with $Y = A f(K,L,Z)$ the marginal productivity of capital is $A \frac{\partial Y}{\partial K}$ and not instead $\frac{\partial Y}{\partial K} = A \frac{\partial f}{\partial K}$.
Obviously the typo is of no consequence since the terms are parameterized for the purpose of estimation.