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OK, say we have some functions, f1, f2, f3, f4, ..., fn. Lets assume that all of these functions take in a real input and give a real output, so we can write y=f1(x), where x,y are both real.

Start with the composition of two functions (to establish notation):

y2 = f2(f1(x)) --> dy2/dx = df2/dx(f1(x)) * df1/dx(x)

in English: "The derivative of y2 with respect to x, evaluated at the point x, is equal to the derivative of f2 with respect to x, evaluated at the point f1(x), times the derivative of f1 with respect to x, evaluated at the point x."

The composition of three functions:

y3 = f3(f2(f1(x))) --> dy3/dx = df3/dx(f2(f1(x))) * df2/dx(f1(x)) * df1/dx(x)

= df3/dx(y2) * dy2/dx

For composition of n functions:

yn = fn(fn-1(...(f2(f1(x)))...))

dyn/dx = dfn/dx(fn-1(...(f2(f1(x)))...)) * ... * df2/dx(f1(x)) * df1/dx(x)

= dfn/dx(fn-1) * dyn-1/dx

Here I used shorthand, so that fn-1 really means f_{n-1}, the "n-1"th function.

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