How to solve this partial derivative which includes a summation?

In summary, the conversation is about a research paper and a specific equation (13) that involves partial differentiation and summation. The question is whether it is correct to take the derivative with respect to Pi when the summation also includes Pi. The solution is provided with a hint that there may be a typo in equation (13). After further discussion, it is confirmed that there was indeed a typo and the solution is corrected.
  • #1
JERRY-thechuha
4
0
I was reading a research paper, and I got stuck at this partial differentiation.

Please check the image which I have uploaded.
check.PNG


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.
 
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  • #2
The partial derivative being with respect to ##P_i##, there is a single term in each sum that depends on ##P_i##.
 
  • #3
ok, I got this.

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

IMG_20170403_175429_HDR.jpg

Anyone please help.
 
  • #4
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##.
 
  • #5
okk, Thanks for the correction.
I also thought that was a typo for log 2.
Please check again the solution.

IMG_20170404_010752_HDR.jpg
 
  • #6
That looks good, taking into account that it is a typo.
 
  • Like
Likes JERRY-thechuha
  • #7
Thanks a lot DrClaude... :-)
 

1. What is a partial derivative?

A partial derivative is a mathematical concept used to calculate the rate of change of a multivariable function with respect to one of its variables while holding the other variables constant. It is denoted by ∂ (the partial derivative symbol) and is often used in fields such as physics, economics, and engineering.

2. What is a summation?

A summation is a mathematical operation that involves adding a sequence of numbers together. It is denoted by the symbol Σ (sigma) and is often used to calculate the total of a set of numbers or to represent a series of terms in a mathematical expression.

3. How do I solve a partial derivative that includes a summation?

To solve a partial derivative that includes a summation, you can use the basic rules of differentiation and apply them to each term in the summation separately. The summation can then be treated as a constant, and the resulting derivatives can be added together to get the final answer.

4. Can a partial derivative be solved using the chain rule?

Yes, the chain rule can be used to solve a partial derivative that involves a composition of functions. In this case, the partial derivative symbol ∂ is used instead of the standard derivative symbol d, and the chain rule is applied to each variable separately.

5. Are there any special cases when solving a partial derivative with a summation?

Yes, there are some special cases when solving a partial derivative with a summation. One example is when the summation involves a constant, in which case the derivative of the constant is 0. Another special case is when the variable of differentiation is not present in one of the terms of the summation, in which case that term will have a derivative of 0.

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