significance of derivate of a function w.r.t other function

Question

I want to calculate the derivative of a function with respect to, not a variable, but respect to another function. What is the significance of it? Suppose $f(x) = \sin(x)$ and $g(x) = \tan(x)$; what is the significance of $\dfrac{\mathrm{d}f(x)}{\mathrm{d}g(x)}$?

Answer 1

You measure the change of the function with respect to the change of the other function.

Recall that the chain rule is given by $$ \frac{df(x)}{dg(x)}\frac{dg(x)}{dx} = \frac{d}{dx}f(g(x)). $$

So you can calculate the derivative of $f(g(x)) $ and divide it by the derivative of $g(x)$ with respect to its argument to obtain what you are looking for. Note that this is not a rigorous derivation.

Another way to do this is to notice that $ dg(x) = g'(x)dx. $ This is called implicit differentiation.

Yet another way for interpreting it is to draw a $2$-dimensional space except the argument axis is replaced with an axis indicating the values of $g(x)$. So what happens when, for instance, $g(x) = x^2$? For $x>1$ you will get to a certain value faster than you would with just $g(x) = x$. So can you see that in that case the graph of your function will be squished?