# Understanding Difference b/w Derivative & Differential in D&K Definition 9.1.3

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In summary, the discussion focuses on the differences between derivatives and differentials in the context of multivariable analysis. The definition of a derivative in the single-variable case is described, where the differential is defined as $dy=f'(x)\,dx$ and is used in integration. Similarly, in the multivariable case, the differential is defined as $df_a(\mathbf{h})=f'(\mathbf{a})\cdot\mathbf{h}$ and is used for sensitivity of dependent variables to changes in independent variables. It is emphasized that a derivative is a function obtained from another function by a limiting process, while a differential is a variable.
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I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ...

I am currently focused on Chapter 9: "Differentiation on $$\displaystyle \mathbb{R}^n$$" ... ...

I need some help with another aspect of Definition 9.1.3 ...

Definition 9.1.3 and the relevant accompanying text read as follows:
https://www.physicsforums.com/attachments/7874
View attachment 7875
In the above text from Junghenn we read the following:

" ... ... The vector $$\displaystyle f'(a)$$ is called the derivative of $$\displaystyle f$$ at $$\displaystyle a$$. The differential of $$\displaystyle f$$ at $$\displaystyle a$$ is the linear transformation $$\displaystyle df_a \in \mathscr{L} ( \mathbb{R}^n, \mathbb{R} )$$ defined by

$$\displaystyle df_a(h) = f'(a) \cdot h, \ \ \ \ \ \ (h \in \mathbb{R}^n )$$ ... ... ... "
My question is as follows:Is the derivative essentially equivalent to the differential ... can we write $$\displaystyle df_a = f'(a)$$ ... if if we can't ... then why not?

... ... indeed, what is the exact difference between the derivative and the differential ...

(I know I have asked a general question like this before ... but this is now in the specific context of Junghenn ...)

Hope someone can help to clarify the above ...

Peter
=========================================================================================***NOTE***

I am aware that the term total derivative and differential are terms used for the same concept ... but this author seems to employ both the term derivative (and Junghenn seems to be defining a total derivative for a scalar function) and differential ...It may also be that the derivative is $$\displaystyle f'(a)$$ and the differential is $$\displaystyle df_a(h) = f'(a) \cdot h$$ ... but then Junghenn states that the differential is $$\displaystyle df_a$$ ... and hence not $$\displaystyle df_a(h)$$ ...Maybe I am making too much of the difference between $$\displaystyle df_a$$ and $$\displaystyle df_a(h)$$ ... ...Peter

Last edited:
Re: Multivariable Analysis ...the derivative and the differential ... D&K Definition 9.1.3 ... ...

Derivatives and differentials are definitely not the same thing. As Berkeley put it in The Analyst, regarding differentials, "May we not call them the ghosts of departed quantities?" In the single-variable case, if you have $y=f(x)$, where $f$ is a differentiable function, then the differential $dx$ is simply a real independent variable, and the differential $dy$ is defined as $dy=f'(x)\,dx$ - a dependent variable. While we often think about differentials as infinitesimals, there's nothing about this definition that requires it. Moreover, this definition has the virtue of corresponding somewhat with the concept of sensitivity of dependent variables to changes in the independent variable. I use differentials in two contexts: integration, where it's simply a notation to A. remind me that I'm integrating w.r.t. a particular variable, and B. assuming we are not using that horrid physicist notation of $\displaystyle\int dx\,f(x)$, it envelopes the integrand, making it abundantly clear exactly what the integrand is.

It's no different in the multivariable case. Instead of $dx$, you have $\mathbf{h}$ - which is just a variable in $\mathbf{R}^n$ - and instead of $dy$, you have $df_a(\mathbf{h})=f'(\mathbf{a})\cdot\mathbf{h}$.

A derivative is a function obtained from another function by a particular limiting process. A differential is a variable, not a function. You definitely can't equate a derivative with a differential. In fact, one rule about differentials is that you can't have a differential on one side of an equation without one on the other side - kind of an unwritten rule, but I find it helps students, particularly when they're doing $u$-substitutions in integration.

Does that help?

Re: Multivariable Analysis ...the derivative and the differential ... D&K Definition 9.1.3 ... ...

Ackbach said:
Derivatives and differentials are definitely not the same thing. As Berkeley put it in The Analyst, regarding differentials, "May we not call them the ghosts of departed quantities?" In the single-variable case, if you have $y=f(x)$, where $f$ is a differentiable function, then the differential $dx$ is simply a real independent variable, and the differential $dy$ is defined as $dy=f'(x)\,dx$ - a dependent variable. While we often think about differentials as infinitesimals, there's nothing about this definition that requires it. Moreover, this definition has the virtue of corresponding somewhat with the concept of sensitivity of dependent variables to changes in the independent variable. I use differentials in two contexts: integration, where it's simply a notation to A. remind me that I'm integrating w.r.t. a particular variable, and B. assuming we are not using that horrid physicist notation of $\displaystyle\int dx\,f(x)$, it envelopes the integrand, making it abundantly clear exactly what the integrand is.

It's no different in the multivariable case. Instead of $dx$, you have $\mathbf{h}$ - which is just a variable in $\mathbf{R}^n$ - and instead of $dy$, you have $df_a(\mathbf{h})=f'(\mathbf{a})\cdot\mathbf{h}$.

A derivative is a function obtained from another function by a particular limiting process. A differential is a variable, not a function. You definitely can't equate a derivative with a differential. In fact, one rule about differentials is that you can't have a differential on one side of an equation without one on the other side - kind of an unwritten rule, but I find it helps students, particularly when they're doing $u$-substitutions in integration.

Does that help?
Thanks for the help Ackbach ...

Still reflecting on what you have written ...

Peter

## 1. What is the difference between a derivative and a differential?

A derivative is a mathematical concept that represents the rate of change of a function at a specific point. It is calculated by finding the slope of the tangent line to the function at that point. A differential, on the other hand, represents the change in a function over a small interval. It is calculated by taking the derivative and multiplying it by the change in the independent variable.

## 2. How are derivatives and differentials related?

Derivatives and differentials are closely related in that the differential is the result of applying the derivative to a function. In other words, the differential is the change in the function over a small interval, while the derivative is the rate of change of the function at a specific point.

## 3. What is the significance of D&K Definition 9.1.3 in understanding the difference between derivatives and differentials?

D&K Definition 9.1.3 provides a formal definition of the derivative and differential, which helps to clarify the difference between the two concepts. It also highlights the relationship between the two and the process of finding the differential using the derivative.

## 4. Can you give an example of how to calculate a derivative and a differential?

Yes, for a function f(x) = x^2, the derivative at x = 3 would be 2x, which is equal to 6. This means that at x = 3, the rate of change of the function is 6. The differential, on the other hand, would be 6 multiplied by the change in x, which could be any small interval, such as 0.1. Therefore, the differential would be 6 * 0.1 = 0.6.

## 5. How can understanding the difference between derivatives and differentials be applied in real-world scenarios?

The concept of derivatives and differentials is essential in various fields, such as economics, physics, and engineering. In economics, derivatives are used to model and predict changes in financial markets. In physics, derivatives are used to calculate velocity and acceleration, and differentials are used to calculate displacement. In engineering, derivatives and differentials are used to design and optimize systems and processes.

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