Finding the derivative in a finite summation

Question

I've the following problem:

$$S=\frac{\partial}{\partial\alpha_1}\left\{\sum_{m=1}^k\left(\alpha_1\cdot x_m^n+\alpha_2\cdot x_m^{n-1}+\dots+\alpha_z\right)^2\right\}$$

How can I reduce this problem to a simpler form?

---

My work, I wrote:

$$S=\sum_{m=1}^k\left\{\frac{\partial}{\partial\alpha_1}\left(\alpha_1\cdot x_m^n+\alpha_2\cdot x_m^{n-1}+\dots+\alpha_z\right)^2\right\}$$

And now I does not know how to go further.

Answer 1

Chain rule $$\frac{\partial}{\partial x}f(x)^2=2f(x)\frac{\partial f}{\partial x}$$ $$S=\sum_{m=1}^k\left\{\frac{\partial}{\partial\alpha_1}\left(\alpha_1\cdot x_m^n+\alpha_2\cdot x_m^{n-1}+\dots+\alpha_z\right)^2\right\}$$ $$S=\sum_{m=1}^k\left\{2 x_m^n \left(\alpha_1\cdot x_m^n+\alpha_2\cdot x_m^{n-1}+\dots+\alpha_z\right)\right\}$$