Chain rule notation - can Leibniz form be made explicit?

In summary: Thank you!In summary, the chain rule can be expressed in two ways, the Leibniz form and the more general form. The Leibniz form is easier to use, but the notations are interchangeable.
  • #1
seand
7
0
Hi there, I'm a new user to the forums (and Calculus) and I 'm hoping you can give me your opinion on my chain rule form below. When learning the chain rule, I was taught two forms. This form:
[tex]\frac{d}{dx}f(g(x))=f'(g(x))g'(x)[/tex]
As well as the Leibniz form
[tex]\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}[/tex] where [tex]y=f(u)[/tex] and [tex]u=g(x)[/tex]

I prefer the Leibniz notation, except that it requires you to understand that [tex]y=f(u)[/tex] and [tex]u=g(x)[/tex], to really understand the d/dx expression.

So my question is if there is a way to make Liebniz more explicit? ie. Does the following make sense? Is it correct?
[tex]\frac{d}{dx}f(g(x)) = \frac{df(g(x))}{dg(x)}\frac{dg(x)}{dx}[/tex]
 
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  • #2
Welcome to PF!

Hi seand! Welcome to PF! :smile:
seand said:
Does the following make sense? Is it correct?
[tex]\frac{d}{dx}f(g(x)) = \frac{df(g(x))}{dg(x)}\frac{dg(x)}{dx}[/tex]

Yes, that's fine.

And I agree, the Leibniz form is easier to use, simply because you can "cancel" in it, just like ordinary fractions. :wink:
 
  • #3
dg/dx = g'(x). The notations are interchangable. Just get used to both of them.
 
  • #4
seand said:
[tex]\frac{d}{dx}f(g(x)) = \frac{df(g(x))}{dg(x)}\frac{dg(x)}{dx}[/tex]
It's possible, but a bit confusing to me. I prefer the non-Leibniz notation anyway. It looks more like (in fact, is) the more general chain rule [tex]\mbox{D}_x\left(g \circ f\right) = \mbox{D}_{f\left(x\right)}\left(g\right) \circ \mbox{D}_x\left(f\right)[/tex].

I find this clearer
[tex]\frac{d}{dx}f(g(x)) = \left.\frac{df(y)}{dy}\right|_{y=g(x)}\frac{dg(x)}{dx}[/tex]
but the nice thing about the expression [tex]\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}[/tex] is that it behaves like fractions, which you lose this way.
 
  • #5
Leibniz notation really prefers that, notationally, you are manipulating algebraically-dependent variables rather than functions.
 
  • #6
Thank you Hurkyl, Landau, mathman and Tiny Tim for the replies and especially the suggestions. It's interesting to know that I could express the chain rule the way I did, but that there are other, more normal ways to do it.
 

FAQ: Chain rule notation - can Leibniz form be made explicit?

What is the chain rule notation?

The chain rule notation is a mathematical concept used in calculus to find the derivative of a composite function. It allows us to find the rate of change of a function composed of two or more functions.

What is Leibniz form in the context of chain rule notation?

Leibniz form is a specific notation used to represent the chain rule in calculus. It uses the notation dy/dx to represent the derivative of a function y with respect to x.

How is Leibniz form made explicit in chain rule notation?

To make Leibniz form explicit, we use the notation d/dx to represent the derivative of a function, and then explicitly state the function inside the parentheses. For example, d/dx (f(g(x))) represents the derivative of the composite function f(g(x)).

Why is the chain rule important in calculus?

The chain rule is important because it allows us to find the derivative of complex functions that are composed of two or more functions. It is a fundamental concept in calculus and is used in many real-world applications, such as physics, engineering, and economics.

Can the chain rule be applied to functions with more than two variables?

Yes, the chain rule can be extended to functions with more than two variables. In this case, we use partial derivatives and the chain rule becomes more complex, but the concept remains the same.

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