Partial derivitive of a summation.

Question

I need some help taking the partial derivative of this function, if it is possible.

Thanks!

Answer 1

You have $\frac{\partial}{\partial x_i} \left(\lambda_1x_1+\lambda_2x_2+\lambda_3x_3+,...,+\lambda_nx_n \right)$

Note that you are taking the partial derivative with respect to a specific $x_i$ and not cycling through all the values of $i$ as you do in the summation $\sum \limits_{i=1}^n$.

For example if $i=2$ you have

$\frac{\partial}{\partial x_2} \left(\lambda_1x_1+\lambda_2x_2+\lambda_3x_3+,...,+\lambda_nx_n \right)$

Or

$0+\lambda_2+0+0+,...+0$

(as $x_1,x_3...x_n$ are all treated as constants)

I have assumed the $\lambda_i$ are just constants and not functions of $x_i$

Answer 2

It will be $\lambda_i$, since all others will be zeros.