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

$$\underline{e_i}(\frac{\partial }{\partial x_i})(\underline{a}\cdot\underline{r})^n=\underline{e_i}(\frac{\partial }{\partial x_i}(a_jx_j)^n)=\underline{e_i}(n(a_jx_j)^{n-1}(a_j\delta_{ij}))$$
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 !