Gradient of dot product using suffix notation

[spacetimedude]

Homework Statement

Find the gradient of \underline{\nabla}(\underline{a}\cdot\underline{r})^n where a is a constant vector, using suffix notation and chain rule.

Homework Equations

On the previous problem,s I found that grad(a.r)=a and grad(r)=\underline{\hat{r}}

The Attempt at a Solution

<br /> \underline{e_i}(\frac{\partial }{\partial x_i})(\underline{a}\cdot\underline{r})^n=\underline{e_i}(\frac{\partial }{\partial x_i}(a_jx_j)^n)=\underline{e_i}(n(a_jx_j)^{n-1}(a_j\delta_{ij}))<br />
I'm sure that the last step is wrong so could someone lead me to the right direction?
Thank you!

 

[HallsofIvy]

The derivative of (ax)^n is NOT n(ax)^{n-1}.

 

[spacetimedude]

The derivative of (ax)^n is NOT n(ax)^{n-1}.

That is what I thought. Could you explain that step?

 

[HallsofIvy]

Just after I wrote that I noticed that you also had "a_j\delta_{ij}". The derivative of (ax)^n= a^nx^n is na^nx^{n-1} but that can also be written as (ax)^{n-2}(a) where the last "a" is due to the chain rule. Was that what you meant?

 

[BvU]

$na^nx^{n-1}$ is confusing. The inner product $(a_j x_j)^{n-1}$ is lost.

When I write it out and then put it back together again $\underline{e_i}(n(a_jx_j)^{n-1}(a_j\delta_{ij}))$ seems OK.
$na_i \underline{e_i}\;(a_jx_j)^{n-1}$ might be somewhat more elegant,
and even that can be simplified further !