This is a beginner question and I want you to help me understand just one step in the following calculus arithmetics. It is taken from my physics book where they want to explain the way to known result that if we have vector function
$$\mathbf b(t)=\beta (t) \cdot \mathbf a(t),$$
where $\beta (t)$ is scalar function and $\mathbf a(t)$ is vector function, then
$$\mathbf b'=\beta'\cdot\mathbf a + \beta\cdot \mathbf a'.$$
First they find increment of $\mathbf b$
$$\Delta \mathbf b=\Delta(\alpha\cdot\mathbf a)=(\alpha + \Delta \alpha)\cdot(\mathbf a+\Delta \mathbf a)-\alpha \cdot \mathbf a$$
then, as usual, the divide it by $\Delta t$ as $\Delta t$ tends to zero. But problem for me arises on the first step where they represent $\Delta(\alpha\cdot \mathbf a)$ as $(\alpha + \Delta \alpha)\cdot(\mathbf a+\Delta \mathbf a)-\alpha \cdot \mathbf a$.
I know that increment of function $y(x)$ is $y(x+\Delta x)-y(x)$. But I dont understand the notation for increment that they use in the book. Please show me why they are equivalent.
If $\Delta y = y(x + \Delta x) - y(x)$ then for an arbitrary product of two functions it holds $\Delta (\alpha * a) = (\alpha * a)(x + \Delta x) - (\alpha * a)(x)$. The Insertion of $(x + \Delta x)$ into $(\alpha * a)$ leads to $(\alpha(x + \Delta x) * a(x + \Delta x))$. Then use the Definition of the increment again.