What are the insights into the Total Derivative formula?

In summary, the Total Derivative of a function is a scalar function that represents how much the original function changes at every point. It is calculated using the multivariable chain rule, with the partial terms indicating how the function changes with respect to a parameter and the full derivatives indicating how the parameters themselves change. The result is a linear operator that maps the differential vector to the differential scalar. The beauty of the differential notation is that the chain rule is built in, allowing for easy calculation of the Total Derivative.
  • #1
quickAndLucky
34
3
I’ve always been confused by the formula for the Total Derivative of a function. $$\frac{df(u,v)}{dx}= \frac{\partial f}{\partial x}+\frac{\partial f }{\partial u}\frac{\mathrm{d}u }{\mathrm{d} x}+\frac{\partial f}{\partial v}\frac{\mathrm{d}v }{\mathrm{d} x}$$
Any insight would be greatly appreciated!

What I do understand:
  1. It’s kind of like a multivariable chain rule
  2. The partials terms tell us how the function changes with respect to a parameter and the full derivatives tell us how the parameters themselves change
  3. The result is a scalar function that represents how much the original function changes at every point

What I don't understand:
  1. If it is a multivariable chain rule then why is it written with partials and not as $$\frac{df(u,v)}{dx}= \frac{\mathrm{d}f }{\mathrm{d} u}\frac{\mathrm{d}u }{\mathrm{d} x}+\frac{\mathrm{d}f }{\mathrm{d} v}\frac{\mathrm{d}v }{\mathrm{d} x}$$
  2. It seems there is no necessary reason for terms to be summed instead of combined in some other functional form (multiplied for example)
  3. What are the conditions that allow us to multiply by the denominator and arrive at the total differential? $$df = \frac{\partial f}{\partial u}du+\frac{\partial f}{\partial v}dv$$Is it really ok to be so cavalier manipulating differentials and derivatives?
  4. Is there a connection between the total derivative and functional derivative that makes the expressions look so similar, or is it just a cosmetic similarity?
 
Physics news on Phys.org
  • #2
I don't agree with your first line. It should read: ## \frac{df(u,v)}{dx}=\frac{\partial{f}}{\partial{u}}\frac{du}{dx}+\frac{\partial{f}}{\partial{v}}\frac{dv}{dx} ##. You only have the first term ## \frac{\partial{f}}{\partial{x}}##, along with the others, if ## f ## has the form ## f(x,u,v) ##.
 
Last edited:
  • #3
Charles Link said:
I don't agree with your first line: ## \frac{df(u,v)}{dx}=\frac{\partial{f}}{\partial{u}}\frac{du}{dx}+\frac{\partial{f}}{\partial{v}}\frac{dv}{dx} ##. You only have the first term if ## f ## has the form ## f(x,u,v) ##.
I think OP is considering ##f[u(x),v(x)]##.
 
  • #4
kuruman said:
I think OP is considering ##f[u(x),v(x)]##.
Yes. I agree. ## u=u(x) ## and ## v=v(x) ##. But unless ## f=f(x,u,v) ## the first ## \frac{\partial{f}}{\partial{x}} ## (for constant ## u ## and ## v ##) does not belong there. ## \\ ## I have seen quite often ## f=f(x,y,z,t) ## for a function describing the density as a function of position and time. Meanwhile if the system is sampled at a location ## (x,y,z) ## that is time dependent, (so that ## x=x(t) ##, ## y=y(t) ## and ## z=z(t) ##), then ## \frac{df}{dt} ## will contain a term ## \frac{\partial{f}}{\partial{t}} ## in addition to terms of the form ## \frac{\partial{f}}{\partial{x}}\frac{dx}{dt} ##, etc.## \\ ## In the function of the OP, he doesn't have this extra parameter in the description of the function. All he has is ## f=f(u,v)=f(u(x),v(x)) ##.
 
Last edited:
  • #5
Yes, you are right, I should have explicitly included a dependence of x in my function.
 
  • Like
Likes Charles Link
  • #6
It is the (possible) occurrence of the explicit x dependency in your original function that makes what you are doing a "Total Derivative" rather than just "a derivative utilizing the multivariable chain rule".

As an insight into the difference between the partial and general derivative notation you can think of "the derivative" of a function of many variables in terms of vectors. I like the Gâteaux definition which boils down to: "A derivative is a linear mapping between differentials":
[tex] dy = f'(x)dx = \lim_{h\to 0} \frac{f(x+h dx)-f(x)}{h}[/tex]
This reads as the derivative is just a multiplier if x and y are scalars, but it becomes a linear operator when x and/or y are vectors. In this context:
[itex]f'(u,v)[/itex] is the operator mapping the differential vector [itex]\langle du, dv\rangle[/itex] to the differential scalar [itex] dy[/itex] as:
[tex] dy = f'(u,v)\bullet \langle du,dv\rangle = \frac{\partial f(u,v)}{\partial u} du + \frac{\partial f(u,v)}{\partial v}dv[/tex]

The beauty of the differential notation and context is that the chain rule is built in. So for the very original context:

[tex] dy = \frac{\partial f(u,v,x)}{\partial u} du + \frac{\partial f(u,v,x)}{\partial v} dv + \frac{\partial f(u,v,x)}{\partial x} dx =[/tex]
[tex] = \frac{\partial f(u,v,x)}{\partial u} \frac{du}{dx} dx+ \frac{\partial f(u,v,x)}{\partial v} \frac{dv}{dx}dx + \frac{\partial f(u,v,x)}{\partial x} dx [/tex]
and thence [tex]dy/dx[/tex] is as stated in the OP.
 
  • Like
Likes Charles Link
  • #7
So we can interpret the partials and differentials differently, thinking of differentials as little vectors and the partials as components of a transformation operator? If this is the right way to think about things, how should I interpret dy/dx? What is the division of 2 vectors?
 

FAQ: What are the insights into the Total Derivative formula?

1. What is a total derivative?

A total derivative is a mathematical concept used in calculus to describe how a function changes with respect to all of its variables. It takes into account the effects of changes in all independent variables on the dependent variable.

2. Why is understanding total derivatives important?

Understanding total derivatives is important in many fields of science and engineering, as it allows us to analyze and predict how different variables affect a system or process. It is particularly useful in optimization, control theory, and economics.

3. How is a total derivative calculated?

A total derivative is calculated by taking the partial derivatives of a function with respect to each of its variables and then summing them together. It can also be expressed using the chain rule or the total differential notation.

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

The main difference between total derivative and partial derivative is that total derivative considers the changes in all independent variables, while partial derivative only looks at the changes in one independent variable at a time. Total derivatives are also used to calculate the slope of a function in multiple dimensions, whereas partial derivatives are used to calculate the slope in one specific direction.

5. How can total derivatives be applied in real-life situations?

Total derivatives have many real-life applications, such as in economics to analyze the relationships between supply and demand, in physics to describe the motion of objects, and in biology to model the growth of populations. They can also be used in engineering to optimize designs and in finance to predict market trends.

Similar threads

Replies
3
Views
2K
Replies
4
Views
2K
Replies
3
Views
3K
Replies
6
Views
1K
Replies
1
Views
2K
Back
Top