Differentiation with respect to a function

In summary, the derivative of a function f(x) with respect to another function g(x) can be found by taking the ratio of their derivatives with respect to the independent variable x.
  • #1
Gabriel Maia
72
1
Hi. A very long problem brought me to a derivative in the form

[itex]\frac{\mathrm{d}f(x)}{\mathrm{d}g(x)}[/itex]


I'm assuming that

[itex]\mathrm{d}g(x)=\left(\frac{\mathrm{d}g(x)}{\mathrm{d}x}\right)\mathrm{d}x[/itex]

So, is it correct to say that

[itex]\frac{\mathrm{d}f(x)}{\mathrm{d}g(x)}=\left(\frac{\mathrm{d}g(x)}{\mathrm{d}x}\right)^{-1}\frac{\mathrm{d}f(x)}{\mathrm{d}x}[/itex]?

Thank you
 
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  • #2
Yes.
(P.S. I don't think "derivating" is a word.....)
 
  • #3
certainly said:
Yes.
(P.S. I don't think "derivating" is a word.....)

differentiation, perhaps?

Thank you very much.
 
  • #4
Your welcome.
It is interesting to note that this does not work with higher derivatives...
[EDIT:- so ##d^2g(x)\neq \Big(\frac{d^2g(x)}{dx^2}\Big) dx^2##]
 
Last edited:
  • #5
Gabriel Maia said:
Hi. A very long problem brought me to a derivative in the form

[itex]\frac{\mathrm{d}f(x)}{\mathrm{d}g(x)}[/itex]


I'm assuming that

[itex]\mathrm{d}g(x)=\left(\frac{\mathrm{d}g(x)}{\mathrm{d}x}\right)\mathrm{d}x[/itex]

So, is it correct to say that

[itex]\frac{\mathrm{d}f(x)}{\mathrm{d}g(x)}=\left(\frac{\mathrm{d}g(x)}{\mathrm{d}x}\right)^{-1}\frac{\mathrm{d}f(x)}{\mathrm{d}x}[/itex]?

Thank you
We have
[tex] \frac{d\, f(x)}{d\, g(x)} = \lim_{\Delta g(x) \to 0} \frac{ \Delta f(x)}{\Delta g(x)}, [/tex]
where
[tex] \Delta f(x) = f(x + \Delta x) - f(x) \doteq f'(x) \Delta x, \\
\Delta g(x) = g(x + \Delta x) - g(x) \doteq g'(x) \Delta x, [/tex]
hence
[tex] \frac{d f(x)} {d g(x)} = \frac{f'(x)}{g'(x)} [/tex]
This is equivalent to what you wrote.
 

1. What is differentiation with respect to a function?

Differentiation with respect to a function is the process of finding the rate of change of a dependent variable with respect to an independent variable. It involves calculating the derivative of the function, which represents the slope of the function at a specific point.

2. Why is differentiation with respect to a function important?

Differentiation with respect to a function is important because it allows us to analyze the behavior of a function and make predictions about its future values. It is also a fundamental concept in calculus and is used in various fields such as physics, engineering, economics, and statistics.

3. What is the difference between differentiation with respect to a function and differentiation with respect to a variable?

Differentiation with respect to a function involves finding the derivative of a function with respect to another function, while differentiation with respect to a variable involves finding the derivative of a function with respect to an independent variable. In other words, the former involves finding the rate of change of one function with respect to another, while the latter involves finding the rate of change of a function with respect to its own variable.

4. What are the rules for differentiation with respect to a function?

The rules for differentiation with respect to a function are similar to the rules for differentiation with respect to a variable. They include the power rule, product rule, quotient rule, chain rule, and sum/difference rule. These rules allow us to find the derivative of more complex functions by breaking them down into simpler parts.

5. Can differentiation with respect to a function be applied to any type of function?

Yes, differentiation with respect to a function can be applied to any type of function, including polynomial, exponential, logarithmic, and trigonometric functions. However, some functions may require more advanced techniques or special rules to differentiate them with respect to another function.

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