I'm trying to create a chain rule formula for multiple composition of functions using the product notation. I want to explain to my students that:
For instance, if we have 4 composition of functions, we can use this chain rule formula:
$H'(x) = (f(g(h(j(x)))))' \cdot (g(h(j(x))))' \cdot (h(j(x)))' \cdot j'(x)$
How do I express this in Pi/product notation $\Pi$? Especially if I want to express it in terms of $...n$ without specifying a number of composition of functions?
Thank you
You could express it as $ f_{1\,.\,.\,n}'(x) = f_1' \left( f_{2\,.\,.\,n}(x) \right) \; f_2' \left( f_{3\,.\,.\,n}(x) \right) \cdots f_{n-1}' \left(f_{n\,.\,.\,n}(x)\right) \; f_n'(x) = \prod_{k=1}^{n} f_k' \left(f_{(k+1\,.\,.\,n)}(x) \right)$
where $f_{i..j}(x) = f_i \circ f_{i+1} ... \circ f_j(x)$