A function m is a composite of functions f and g if
m(x) = f[g(x)]
The domain of m is the set of all numbers x such that x is in the domain of g, and g(x) is in the domain of f.
If u(x) is a differentiable function, n is any real number, and
y = f(x) = [u(x)]n
f'(x) = n[u(x)]n-1u'(x)
Using simplified notation,
y' = nun-1u'ord/dx un = nun-1du/dxwhereu = u(x)
If m(x) = E[I(x)] is a composite function, then
m'(x) = E'[I(x)]I'(x)
provided that E'[I(x)] and I'(x) exist. Equivalently, if y = E(u) and u = I(x), then
dy/dx = dy/du du/dx
provided that dy/du and du/dx exist.