- #1

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## Summary:

- Total Derivative

## Main Question or Discussion Point

If we have two functions C(y(t), r(t)) and I(y(t), r(t)) can we write $$\frac{\frac{dC}{dt}}{\frac{dI}{dt}}=\frac{dC}{dI}$$?

- I
- Thread starter Ahmed Mehedi
- Start date

- #1

- 39

- 5

## Summary:

- Total Derivative

If we have two functions C(y(t), r(t)) and I(y(t), r(t)) can we write $$\frac{\frac{dC}{dt}}{\frac{dI}{dt}}=\frac{dC}{dI}$$?

- #2

- #3

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Essentially yes, but you need to be careful that it all makes sense. In this case we can define:Summary::Total Derivative

If we have two functions C(y(t), r(t)) and I(y(t), r(t)) can we write $$\frac{\frac{dC}{dt}}{\frac{dI}{dt}}=\frac{dC}{dI}$$?

$$f(t) = C(y(t), r(t)) \ \ \text{and} \ \ u(t) = I(y(t), r(t))$$

Then ##\frac{df}{dt}## and ##\frac{du}{dt}## are well defined. You also have to imagine that you express ##t## as a function of ##u##, so that we have a further function:

$$F(u) = f(t(u))$$

Then:

$$\frac{dF}{du} = \frac{df}{dt} \frac{dt}{du} = \frac{df/dt}{du/dt}$$

- #4

- 39

- 5

Thanks a lot! You have been very helpful!Essentially yes, but you need to be careful that it all makes sense. In this case we can define:

$$f(t) = C(y(t), r(t)) \ \ \text{and} \ \ u(t) = I(y(t), r(t))$$

Then ##\frac{df}{dt}## and ##\frac{du}{dt}## are well defined. You also have to imagine that you express ##t## as a function of ##u##, so that we have a further function:

$$F(u) = f(t(u))$$

Then:

$$\frac{dF}{du} = \frac{df}{dt} \frac{dt}{du} = \frac{df/dt}{du/dt}$$

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