Chain rule notation - can Leibniz form be made explicit?

[seand]

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:
$$\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$$
As well as the Leibniz form
$$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$$ where $$y=f(u)$$ and $$u=g(x)$$

I prefer the Leibniz notation, except that it requires you to understand that $$y=f(u)$$ and $$u=g(x)$$, 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?
$$\frac{d}{dx}f(g(x)) = \frac{df(g(x))}{dg(x)}\frac{dg(x)}{dx}$$

 

[tiny-tim]

Welcome to PF!

Hi seand! Welcome to PF!

Does the following make sense? Is it correct?
$$\frac{d}{dx}f(g(x)) = \frac{df(g(x))}{dg(x)}\frac{dg(x)}{dx}$$

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.

 

[mathman]

dg/dx = g'(x). The notations are interchangable. Just get used to both of them.

 

[Landau]

$$\frac{d}{dx}f(g(x)) = \frac{df(g(x))}{dg(x)}\frac{dg(x)}{dx}$$

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 $$\mbox{D}_x\left(g \circ f\right) = \mbox{D}_{f\left(x\right)}\left(g\right) \circ \mbox{D}_x\left(f\right)$$.

I find this clearer
$$\frac{d}{dx}f(g(x)) = \left.\frac{df(y)}{dy}\right|_{y=g(x)}\frac{dg(x)}{dx}$$
but the nice thing about the expression $$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$$ is that it behaves like fractions, which you lose this way.

 

[Hurkyl]

Leibniz notation really prefers that, notationally, you are manipulating algebraically-dependent variables rather than functions.

 

[seand]

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.