The following is taken from a wiki article on Brilliant - web address: https://brilliant.org/wiki/lagrangian-formulation-of-mechanics/
The article is on the Langrangian and its derivation. During the derivation the following equation is given as a statement of D'Alambert's principle:
$$\vec{F}_{\text{net}} \cdot \delta \vec{r} = m \frac{d^2\vec{r}}{dt^2}\cdot \delta \vec{r}$$
After which the following equation is used: $(xy)' = x'y + xy'$ to rewrite the right hand side so that:
$$m \frac{d^2\vec{r}}{dt^2}\cdot \delta \vec{r} = m \left[\frac{d}{dt}\left(\frac{d\vec{r}}{dt}\cdot\delta\vec{r}\right) - \frac{d\vec{r}}{dt} \cdot \frac{d\delta\vec{r}}{dt}\right] $$
What is $x$, what is $y$ and how is the equation applied?
$x$ and $y$ are some functions of variable $t$. The derivative is with respect to this variable, so $x'$ is equivalent to $\frac{dx}{dt}$. The formula for $(xy)'$ is the derivative of a product of functions. In this case they use $$x=\frac{d\vec r}{dt}$$ and $$y=\delta\vec r$$ Then $$\frac{d}{dt}\left(\frac{d\vec r}{dt}\delta\vec r\right)=\left(\frac{d}{dt}\frac{d\vec r}{dt}\right)\delta\vec r+\frac{d\vec r}{dt}\frac{d}{dt}\delta\vec r$$ They just move the last term to the other side