Differentiating the distance formula with respect to time

Question

How does one apply implicit differentiation to the distance formula with respect to time $t$?

$$d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$$

Answer 1

Let introduce

$$ X(t)=(x_1(t)-x_2(t))^2+(y_1(t)-y_2(t))^2$$

Differentiating relatively to $t$:

$$X'(t)=2[(x_1'(t)-x_2'(t))(x_1-x_2)+(y_1'(t)-y_2'(t))(y_1(t)-y_2(t))] $$

So :

$$ d'(t)=\dfrac{X'(t)}{2\sqrt{{X}(t)}}=\dfrac{2[(x_1'(t)-x_2'(t))(x_1(t)-x_2(t))+(y_1'(t)-y_2'(t))(y_1(t)-y_2(t))}{2d(t)} $$

Is that convenient ? If you need more explaination don't hesitate.