Does the graph of a function $f(x)$ versus its derivative $f'(x)$ have a special purpose?

Question

I can't find any insights online on how useful the graph of function $f(x)$ (on the $y$ axis) versus it's derivative $f'(x)$? (on the $x$ axis) does it provide some useful informations if any?

For example, I see that when I plot $\sin(x)$ against $\cos(x)$ the plot is a circle which is reminiscent of parametric equations.

My question is: Assuming the function is nice (nice in the common sense that it is continuous) does that kind of graph provide any useful information? What about this graph where $x(t) = f'(t)$:

f(t) versus x(t) where x(t) = f'(t)

Answer 1

There is this really good popular video from MinutePhysics titled How To Tell If We're Beating COVID-19.

It plots "total number of cases" against "rate of change" for different countries and over time.

Even if you are comlpetely annoyed by pandemic stuff by now (and the video is 1 1/2 years old and from March '20), it's still interesting to see how different graphing can give more insight into a topic.

Answer 2

You can use it to analyze the behavior of differential equations. For example, consider $$y' = y^2 - 5x+6$$ Plotting $y'$ vs $y$ shows us the zeros (constant solutions/asymptotes for non constant solutions) and places where y' is positive or negative. This gives us the following qualitative behaviors that are useful to understand

$$y(0)<2 \implies \begin{cases}y'(x)>0\\\lim\limits_{x\to-\infty}y(x) = -\infty\\\lim\limits_{x\to+\infty}y(x)=2\end{cases}$$

$$y(0)=2 \implies y(x)=2$$

$$2<y(0)<3 \implies \begin{cases}y'(x)<0\\\lim\limits_{x\to-\infty}y(x) = 3\\\lim\limits_{x\to+\infty}y(x)=2\end{cases}$$

$$y(0)=3 \implies y(x)=3$$

$$y(0)>3 \implies \begin{cases}y'(x)>0\\\lim\limits_{x\to-\infty}y(x) = 3\\\lim\limits_{x\to+\infty}y(x)=+\infty\end{cases}$$

All without going through the trouble of solving the differential equation. This qualitative theory of ODEs is useful in analyzing more complex dynamics, especially when analytic solutions are intractable and precise numerics are unnecessary.

Answer 3

Such a graph treats $x$ as a parameter.

In dynamic systems ( vibrations/ oscillations etc. ) plots $ (f(t), f'(t))$ with time $t$ as parameter are called phase portraits; they are very useful, as they help to represent non-linearities of oscillatory phenomena through derivatives of motion e.g., in Van der Pol equation and Limit cycles.. to include effects of damping. Another example id population growth dynamics.

In coupled differential equations with several state variables phase portrait for each state variable $\{(x(t),x'(t)),(y(t),y'(t))\}$ can be plotted separately to study.