How to solve this partial derivative which includes a summation?

[JERRY-thechuha]

I was reading a research paper, and I got stuck at this partial differentiation.

Please check the image which I have uploaded.

Now, I got stuck at Equation (13).
How partial derivative was done, where does summation gone?
Is it ok to do derivative wrt Pi where summation also includes Pi?

Please provide me solution or any reference. Its very urgent.

 

[DrClaude]

The partial derivative being with respect to $P_i$, there is a single term in each sum that depends on $P_i$.

 

[JERRY-thechuha]

ok, I got this.

Please tell me is this correct way to solve this question?

Anyone please help.

 

[DrClaude]

Your first derivative is correct, but not your second.

Looking at the original text, I suspect that there is a typo in eq. (13). What is written as
$$
\log_2 \left(\frac{P_i k_i}{B_i N_0} + 1 \right)^2
$$
in the denominator should read
$$
\log(2) \left(\frac{P_i k_i}{B_i N_0} + 1 \right)^2
$$
The same goes for the other $\log_2$.

 

[JERRY-thechuha]

okk, Thanks for the correction.
I also thought that was a typo for log 2.
Please check again the solution.

 

[DrClaude]

That looks good, taking into account that it is a typo.

 

[JERRY-thechuha]

Thanks a lot DrClaude... :-)