For my personal interest I am working my way through Strogatz's Non-linear Dynamics and Chaos book in order to better understand the background. The problem I am doing is filling in the steps (and why) for the second equation to deriving the perturbation/averaging equations for $\ddot{x} + x + \epsilon h(x,\dot{x})= 0$. This also is covered in https://youtu.be/ma6wOguLxnI It makes sense that:
$\dot{x} = y$
$\dot{y} = -x - \epsilon h(x,\dot{x})$
I understand that solutions, when $\epsilon = 0$, are like:
$$x(t) = r cos(t + \phi{(t)})$$ and
$$y(t) = -r sin(t + \phi{(t)})$$
Deriving $\dot{r} = \epsilon h(x,\dot{x})sin(t + \phi{(t)})$ was clear. However, in the youtube clip, it was stated that:
$$\frac{d}{dt}(t + \phi(t)) = \frac{d}{dt}\Bigg(tan^{-1}\bigg(\frac{-y(t)}{x(t)}\bigg)\Bigg) \ (1)$$ I understand that this becomes the expression:
$$\dot{\phi} = \frac{\epsilon h(x,\dot{x})}{r(t)}cos(t + \phi{(t)})\ (2)$$
After several attempts at reasonable substitutions, I have managed that: $$\frac{d}{dt}(t + \phi(t)) = \frac{d}{dt}\Bigg(tan^{-1}\bigg(\frac{-y(t)}{x(t)}\bigg)\Bigg)$$ $$\frac{d}{dt}(t + \phi(t)) = \frac{-x\dot{y} + y\dot{x}}{x^2 + y^2}$$ $$\frac{d}{dt}(t + \phi(t)) = \frac{-x(-x -\epsilon h(x,y)) + y^2}{x^2 + y^2}$$ $$\frac{d}{dt}(t + \phi(t)) = \frac{x^2 +y^2 + \epsilon h(x,y) x}{x^2 + y^2}$$ $$\frac{d}{dt}(t + \phi(t)) = 1 + \frac{\epsilon h(x,y) x}{r^2}$$ $$\frac{d}{dt}(t + \phi(t)) = 1 + \frac{\epsilon h(x,y) r cos(t + \phi(t))}{r^2}$$ $$\frac{d}{dt}(t + \phi(t)) = 1 + \frac{\epsilon h(x,y) cos(t + \phi(t))}{r}$$ Which now, thanks @Triatticus cancels out the 1 on both sides.
$$\dot{\phi(t)} = \frac{\epsilon h(x,y) cos(t + \phi(t))}{r}$$
In writing out this question, the algebra became more clear. What I don't understand is the reasoning why $\frac{d}{dt}(t + \phi(t)) = \dot{\phi}$. Where does the $t$ go? Ought it not be $\frac{d}{dt}(t + \phi(t)) = \dot{\phi} + 1$$? I don't want to mindlessly apply something that I don't understand fully... If anyone can explain the specific question's answer and maybe point me to some more readable material as a background, I would be most appreciative.