# How do i perform differentiation over summation?

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• mahmud_dbm
In summary, there is confusion over the meaning of ##\frac{d}{dx} y##. It is not clear if it is differentiating a vector ##y## with respect to another vector ##x##, or a vector of functions of the scalar variable ##x## with respect to ##x##. It is also unclear how to define this sort of derivative. Additionally, there is a question of how to differentiate a vector of constants with respect to another vector. However, if there is a function ##f(x)## and a set of constants ##c_1, c_2,...##, it is possible to form the derivatives ##f'(c_1), f'(c_2),...##.
mahmud_dbm

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It isn't clear what you mean by ##\frac{d}{dx} y##. Are you differentiating a vector ##y## with respect to to another vector ##x## ? (If so, how do you define that sort of derivative?) Or are you differentiating a vector of functions of the scalar variable ##x## with respect to ##x## ?

You say that ##x## has exactly N samples. If ##x## is vector of constants, I don't what you would mean by differentiating with respect to ##x##. If you have some function ##f(x)## and some set of constants ## c_1,c_2,...## you can talk about forming the derivatives ## f'(c_1), f'(c_2),...##.

You are right, here x is a vector and H is a matrix, and i want to differentiate Hx with respect to x

Here's the complete problem

that's how far i have gone, probably it's wrong.

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Stephen Tashi said:
It isn't clear what you mean by ##\frac{d}{dx} y##. Are you differentiating a vector ##y## with respect to to another vector ##x## ? (If so, how do you define that sort of derivative?) Or are you differentiating a vector of functions of the scalar variable ##x## with respect to ##x## ?

You say that ##x## has exactly N samples. If ##x## is vector of constants, I don't what you would mean by differentiating with respect to ##x##. If you have some function ##f(x)## and some set of constants ## c_1,c_2,...## you can talk about forming the derivatives ## f'(c_1), f'(c_2),...##.

please let me know if it's okay!
that's how far i have gone, probably it's wrong.

mahmud_dbm said:
You are right, here x is a vector and H is a matrix, and i want to differentiate Hx with respect to x.

You still haven't explained what ##x## is.

## What is differentiation over summation?

Differentiation over summation is a mathematical operation that involves finding the derivative of a summation. It is commonly used in calculus to find the rate of change of a summation function.

## What are the basic rules for differentiating over summation?

There are several rules for differentiating over summation, including the linearity rule, the power rule, and the chain rule. These rules are similar to the rules for differentiating regular functions, but they are applied to each term in the summation.

## Can all summation functions be differentiated?

No, not all summation functions can be differentiated. Some summation functions are not defined for all values of the variable and therefore cannot be differentiated. Additionally, some summation functions may be too complex to be differentiated analytically.

## How do I differentiate a summation function with multiple variables?

Differentiating a summation function with multiple variables involves applying the same rules as differentiating a single variable summation function. However, the partial derivatives of each term in the summation should be taken with respect to the corresponding variable.

## Can differentiation over summation be used in real-world applications?

Yes, differentiation over summation is commonly used in many real-world applications, such as in physics, economics, and engineering. It allows us to calculate rates of change and optimize functions in various fields.

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