# Partial derivative - why is it count like this?

• player1_1_1
In summary, The conversation discusses the use of a function composition derivative to evaluate an integral in classical mechanics. The underintegral function, f, is dependent on y, y', and x, and when y is replaced with a function of alpha, the chain rule must be applied. The person in the conversation is struggling to understand why both y and y' are involved in the derivative when it is only with respect to alpha. They suggest using a non-linear expression for f to gain a better understanding.
player1_1_1

## Homework Statement

I have John R Taylor "Classical mechanics" part 1, and I have an integral:
$$\int\limits^{x_2}_{x_1}f\left(y+\alpha\eta,y^{\prime}+\alpha\eta^{\prime},x\right)\mbox{d}x$$
and here is count derivative of underintegral function in $$\alpha$$
$$\frac{\partial f\left(y+\alpha\eta,y^{\prime}+\alpha\eta^{\prime},x\right)}{\partial\alpha}=\eta\frac{\partial f}{\partial y}+\eta^{\prime}\frac{\partial f}{\partial y^{\prime}}$$
why there is suddenly $$\frac{\partial f}{\partial y^{\prime}}$$ and $$\frac{\partial f}{\partial y}$$, while this derivative is by $$\alpha$$ - why not only $$\eta,\eta^{\prime}$$?

## Homework Equations

I was thinking about function composition derivative, but it didnt helped me.

## The Attempt at a Solution

Nothing, I couldn't do anything with this, I don't know why this is count like this, please help;] thanks!

$$f=f(y,y',x)$$

Now if you replace $$y=\bar{y}+\alpha\eta$$ then y becomes a function of alpha, which means that you have to use the chain rule.

Try choosing an explicit non-linear expression for f if that makes it more clear. For example f(y,y',x) = y2 + y'2

## 1. What is the purpose of partial derivatives?

Partial derivatives are used to measure how a function changes with respect to one specific variable, while holding all other variables constant. They are particularly useful in multivariable calculus and are essential for understanding the rate of change in complex systems.

## 2. Why are partial derivatives counted differently than regular derivatives?

Partial derivatives are counted differently because they measure the change in a function with respect to one specific variable, unlike regular derivatives which measure the overall change of a function. This is due to the fact that in multivariable functions, there are multiple independent variables that can affect the output, so the partial derivative focuses on one variable at a time.

## 3. How do you calculate a partial derivative?

To calculate a partial derivative, you differentiate the function with respect to one variable, treating all other variables as constants. This is denoted by using the partial derivative symbol (∂) instead of the regular derivative symbol (d). The resulting partial derivative will be a function of the variable that you differentiated with respect to.

## 4. What is the difference between a partial derivative and a total derivative?

The main difference between a partial derivative and a total derivative is that a partial derivative measures the change in a function with respect to one specific variable, while a total derivative measures the overall change of a function as a result of changes in all variables. A total derivative takes into account the interactions between all variables, while a partial derivative only focuses on one variable at a time.

## 5. In what fields is the concept of partial derivatives commonly used?

Partial derivatives are commonly used in the fields of physics, engineering, economics, and statistics. They are particularly useful in studying systems with multiple variables and understanding the rate of change in complex systems. They are also important in optimization problems, where finding the maximum or minimum of a multivariable function requires the use of partial derivatives.

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