I have a doubt about the Leibniz's notation in chain rule.
Suppose that $f(x) = \tan^n(x)$.
I want to use the Leibniz's notation, so I think that I will have:
Let ${u(x)=\tan(x)}$
$${\frac{d}{dx}f(x)}=\frac{d}{du}u(x)^n\cdot \frac{d}{dx}u(x).$$
Other examples(that i think is good)
let ${m(x)=2x}$
$${\frac{d}{dx}\sin(2x)= \frac{d}{dm}\sin(m(x))\cdot \frac{d}{dx}m(x)}.$$
Is my use of Leibniz's notation correct?
Some classic books write following: if $z=F(y)$ and $y=f(x)$, then in some appropriate conditions we can write $$\frac{dz}{dx} = \frac{dz}{dy}\frac{dy}{dx}$$. Some think, that more good is $$\frac{dF \circ f}{dx} = \frac{dF}{dy}\frac{df}{dx}$$ Both do not use function in denumerator. So for your first example more conventionally will be to write $z=F(y)=y^n$ and $y=f(x)=\tan x$, so $$\frac{dz}{dx} = \frac{d \tan^nx}{dx} = \frac{dz}{dy}\frac{dy}{dx}=\frac{dy^n}{dy}\frac{d \tan x}{dx}$$ Same for second.