Generalization of Chain Rule for n Composite Functions

Question

Suppose you have a function $f$ such that $$ f(x) = a(b(c(d(x)))) $$

Where $a$, $b$, $c$, and $d$ are all differentiable functions. You're asked to find $f'(x)$.

My question is this - Can you "chain" the Chain Rule for 4 functions and state that $$ f'(x) = a'(b(c(d(x)))) *b'(c(d(x)))*c'(d(x))*d'(x) $$

And even more generally, can you apply the same pattern for $n$ functions? I searched three books on this and didn't find a general result.

Answer 1

With $f=a \circ b \circ c \circ d$ you have using chain rule: $$f'(x)=(a\circ (b \circ c \circ d))'(x)=a'((b \circ c \circ d)(x)) \times (b \circ c \circ d)'(x)$$ using once again chain rule: $$f'(x)=a'(b(c(d(x)))) \times (b \circ (c \circ d))'(x)=a'(b(c(d(x)))) \times \left[b' \left(( c \circ d)(x)\right) \times( c \circ d)'(x) \right]$$ and using one last time the chain rule on $c \circ d$ you obtain the result.

Remark: you can show that the same pattern hold for $n$ function by induction over $n$.

Answer 2

Yes you can inductively apply the chain rule for 4 or more functions.