This is from Mathematical Methods in the Physical Sciences, Mary L. Boas 3rd edition, Chapter 4, section 11, page 240. During a derivation of the Laplace equation in polar coordinates using abbreviations such as $G = \frac{\partial F}{\partial x}$ and $H = \frac{\partial F}{\partial y}$ and the substituting it in and taking its partial derivative, I do not see how we do not have to apply the product rule. So getting to the exact part which confuses me, we have,
$$\frac{\partial F}{\partial x} = \frac{\partial F}{\partial r} \frac{\partial r}{\partial x} + \frac{\partial F}{\partial \theta} \frac{\partial \theta}{\partial x}= \cos(\theta) \frac{\partial F}{\partial r} - \frac{\sin(\theta)}{r}\frac{\partial F}{\partial \theta}$$
$$\frac{\partial F}{\partial y} = \frac{\partial F}{\partial r} \frac{\partial r}{\partial y} + \frac{\partial F}{\partial \theta} \frac{\partial \theta}{\partial y}= \sin(\theta) \frac{\partial F}{\partial r} + \frac{\cos(\theta)}{r}\frac{\partial F}{\partial \theta}$$
Then we let $G = \frac{\partial F}{\partial x}$ and $H = \frac{\partial F}{\partial y}$ thus,
$$G = \cos(\theta) \frac{\partial F}{\partial r} - \frac{\sin(\theta)}{r}\frac{\partial F}{\partial \theta}$$ $$H = \sin(\theta) \frac{\partial F}{\partial r} + \frac{\cos(\theta)}{r}\frac{\partial F}{\partial \theta}$$
This is the part where I become confused on, which is simply letting,
$$\frac{\partial G}{\partial x} = \cos(\theta) \frac{\partial G}{\partial r} - \frac{\sin(\theta)}{r}\frac{\partial G}{\partial \theta} \tag{3}$$
$$\frac{\partial H}{\partial y} = \sin(\theta) \frac{\partial H}{\partial r} + \frac{\cos(\theta)}{r}\frac{\partial H}{\partial \theta} \tag{4}$$
So this is the part that confuses me, as for example $\frac{\partial G}{\partial x} = \frac{\partial^2 F}{\partial x^2} = \frac{\partial}{\partial x} \left(\cos(\theta) \frac{\partial F}{\partial r} - \frac{\sin(\theta)}{r}\frac{\partial F}{\partial \theta} \right)$. $\cos(\theta)$ can be represented as a function of x such as $\frac{x}{r}$ So how can we just bypass differentiating these parts by simply replacing it with G and then getting equations 3 and 4? I am new to manipulating derivatives in this manner and I am currently teaching myself, any help would be great! Thank you again!