# Derivative of the cross and dot product

1. Jan 27, 2008

### bluelagoon

1. The problem statement, all variables and given/known data

If you have two functions dependent on t, A(t) and B(t). Prove their derivatives are as follows:

d(A (dot) B) / dt = [A (dot) (dB)/(d(t)] + [d(A)/d(t) (dot) B]

{Where "(dot)" acts as the dot product}

d(A x B) / dt = [A x (dB)/(d(t)] + [d(A)/d(t) x B]

{Where "x" acts as the cross product}

I have little experience with differentiating cross products and dot products and appreciate any help in starting this one!

Thanks,
Janet

2. Jan 27, 2008

### nicksauce

Well A . B = axbx + ayby + azbz (in 3 dimensions)
Differentiate that using the product rule, then regroup and show that its the expression you need it to be equal to. The cross product is similar.

3. Jan 27, 2008

### Biest

I think the easy way around of using $$\delta_{ij}$$ and $$\epsilon_{ijk}$$ is a bit too advanced.

For the cross-product I would just say from experience that working backward may be a bit less confusing, e.g. show what two parts are and simplify the output expression, and then showing that it is equal to the derivative of the initial expression

4. Jan 27, 2008

### Hurkyl

Staff Emeritus
You've already proven the derivative rule for one kind of product; doesn't the same method work for these products?