The Ratio of Total Derivatives

[Ahmed Mehedi]

TL;DR

Total Derivative

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

 

[Ahmed Mehedi]

@dRic2 @PeroK @etotheipi

 

[PeroK]

Summary:: Total Derivative

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

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

 

[Ahmed Mehedi]

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

Thanks a lot! You have been very helpful!