Solution to Related Functions

let:f(x) = g(x+1)-x, g(x) = f(2x)-2

Now, we want to find the derivative of f( g(x) ). It is important to remember that these functions are related, entangled, if you will. With that in mind, we must not forget that derivates do not require precise functions. So. f(x) = g(x+1) – x, and that equals f(2x+2) – 2. That makes f(x) = f(2x+2) – x – 2. That means when we add x+2 to x, f(x) increases by x+2. Therefore the derivative of f(x) is 1. Now, g(x) is 2 less than twice f(x), and thus its derivative is the same as that of 2f(x). That makes g(x)’s derivative 2. Because the derivative of f( g(x) ) equals their separate derivates multiplied. That makes our answer 2.

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