# The Ratio of Total Derivatives

• I

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 $$\frac{\frac{dC}{dt}}{\frac{dI}{dt}}=\frac{dC}{dI}$$?

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

• dRic2, etotheipi, Math_QED and 1 other person
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!