Can Derivatives be Taken with Respect to Functions?

[Char. Limit]

Is it possible to take a derivative with respect to a function, rather than just a variable? I'll give a simple example of how I imagine such a thing would work to try to explain...

$$\frac{d}{d(sin(x))}\left(sin^2(x)\right) = 2 sin(x)$$

Can you take a derivative this way?

Also, can you write an equivalent integral as such?

$$\int 2 sin(x) d(sin(x)) = sin^2(x) + C$$

 

[Bohrok]

I was thinking the same thing after looking at https://www.physicsforums.com/showthread.php?t=422688".

I found this site about differentiating with respect to functions:
http://www.transtutors.com/calculus-homework-help/differentiation/differentiation-with-respect-to-another-function.aspx
Looks quite easy just using the chain rule.

As far as integrating, no clue so far.

 

[alxm]

Sounds like you're looking for http://en.wikipedia.org/wiki/Functional_derivative" .

 

[ross_tang]

To differentiate f(x) w.r.t g(x), just do the following:

$$\frac{d f(x)}{d g(x)}$$

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

$$=\frac{d f(x)}{d x}\left(\frac{d g(x)}{d x}\right)^{-1}$$

Char. Limit, for the integration, what you have written is the standard way I learned to do integration. The general form is:

$$\int g(x) f'(x)d x=\int g(x)d f(x)$$

For example, by using the property of differential, $$d x^2/2 = x d x$$

$$\int e^{x^2}x \text{dx}$$

$$=\frac{1}{2}\int e^{x^2}\text{dx}^2$$

$$=\frac{1}{2}e^{x^2}+C$$

This method of doing integration is much better than using substitution in many situations. Since if the integral is complicated, you don't have to do substitution repeatedly.

 

[Dickfore]

Sounds like you're looking for http://en.wikipedia.org/wiki/Functional_derivative" .

no, it is just parametric ordinary differentiation.

$$w = f(u) = u^{2}, u(x) = \sin{(x)} \Rightarrow \frac{dw}{du} = f'(u) = 2 u = 2 \sin{(x)}$$

Functional derivatives are derivatives of functionals with respect to functions. $\sin^{2}{(x)}$ is not a functional.

 

[alxm]

Functional derivatives are derivatives of functionals with respect to functions. $\sin^{2}{(x)}$ is not a functional.

Ah yes, silly me.. Guess I've been reading too many DFT papers lately.

 

[Pengwuino]

You'll even see things such as integrals over volumes, $$d^3x$$, done in terms of $$d(cos(\theta))$$ instead of just $$d\theta$$. So instead of $$r^2sin(\theta)drd\theta d\phi$$, you can change your limits of integration to be $$r^2drd(cos(\theta))d\phi$$. My first encounter, and I bet a lot of physics majors first encounter, is integrating Legendre Polynomials that are dependent on $$cos(\theta)$$ so it is only natural.

 

[Dickfore]

It's basically the method of substitution "on the go".

 

[HallsofIvy]

It's really just the chain rule. The "derivative of f(x) with respect to g(x)" is
$$\frac{df}{dg}= \frac{df}{dx}\frac{dx}{df}= \frac{\frac{df}{ex}}{\frac{dg}{dx}}$$.
It is simply "the rate of change of f compared to that of g" or "the rate of change of f divided by the rate of change of g".

And, as others have said, integration in that form is just "substitution" which is, itself, the "chain rule in reverse".