I'm learning multiple applications of the chain rule and the notation surrounding it. Does the following notation with example correctly reflect the chain rule in both Lagrange and Leibniz notation?
The derivative: $$ y = f(x) $$ $$ \frac {dy}{dx} = f'(x) = \frac {dy}{dx} \Big|_{x} = \lim_{h \to 0} {\frac {f(h+x) - f(x)}{(h+x) -x}} = \lim_{x_1 \to x} {\frac {f(x_1) - f(x)}{x_1 -x}} $$ or when: $$ x=a, \frac {dy}{dx} = f'(a) $$
The chain rule: $$ y = f(u), u =g(x) $$ $$ \frac {dy}{du} \Big|_{g(x)} \cdot \frac {du}{dx} \Big|_{x} = (f \circ g)'(x) = (f' \circ g)(x) \cdot g'(x) $$
The chain rule applied two times: $$ f(x) = (f_1 \circ (f_2 \circ f_3))(x) $$
$$ \begin{align*} f'(x) &= (f_1 \circ (f_2 \circ f_3))'(x) = (f_1' \circ (f_2 \circ f_3))(x) \cdot (f_2 \circ f_3)'(x) \\ &= (f_1' \circ (f_2 \circ f_3))(x) \cdot (f_2' \circ f_3)(x) \cdot f_3'(x) \end{align*} $$
Decompose composite into separate functions: $$ f'(x) = \frac {d f} {dx} = \frac {dh} {dj} \frac {dj} {dk} \frac {dk} {dx} $$ $$ f = h = f_1(j), j = f_2(k), k = f_3(x) $$
$$ f'(x) = \frac {d f} {dx} \Big|_{x} = \frac {dh} {dj} \Big|_{f_2(f_3(x))}\frac {dj} {dk} \Big|_{f_3(x)} \frac {dk} {dx} \Big|_{x} $$
which agrees with above.