# Verify derivative of a dot product.

Let $$\vec w(t) \;,\; \vec v (t)$$ be 3 space vectors that is a function of time t. I want to verify that:

$$\frac {d(\vec w \cdot \vec v)}{dt} = \vec v \cdot \frac { d\vec w}{dt} \;+\; \vec w \cdot \frac { d\vec v}{dt}$$

I work through the verification by splitting w and v into x, y, z components, do the dot product and take the derivative to verify already. Just want to run this by the expert to confirm.

Thanks

Alan

## Answers and Replies

arildno
Homework Helper
Gold Member
Dearly Missed
That will work.

Note, however, that by definition, the dot product has the distributive property of multiplication:
$$(u+du)\cdot(v+dv)=u\cdot{v}+u\cdot{dv}+v\cdot{du}+du\cdot{dv}$$
For all vectors u,du,v and dv.

Thus, the result for the derivative ought to be apparent..

That will work.

Note, however, that by definition, the dot product has the distributive property of multiplication:
$$(u+du)\cdot(v+dv)=u\cdot{v}+u\cdot{dv}+v\cdot{du}+du\cdot{dv}$$
For all vectors u,du,v and dv.

Thus, the result for the derivative ought to be apparent..
Thanks

But I don't see how

$$(u+du)\cdot(v+dv)=u\cdot{v}+u\cdot{dv}+v\cdot{du}+du\cdot{dv}$$

relate to my original question. Please explain.

Thanks

Alan

arildno