My textbook, Advanced Macroeconomics (Romer), introduces a function $\displaystyle Y=F(K,AL)$, which can be written $\displaystyle \frac{Y}{AL}=F(\frac{K}{AL},1)$ and is often shortened to $\displaystyle y=f(k)$. It then suggests that $\displaystyle \frac{\partial F(K,AL)}{\partial K} = ALf'(\frac{K}{AL})(\frac{1}{AL})$.

But how can they take the partial with respect to K if they don't specify the actual function? They say what the function depends on, but not what the actual function is. What is happening here?