Nested Squareroot

Calculus Level 2

Consider the function

f(x)=x+x+x+. f(x) = \sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}.

If the gradient of the tangent of the function at x=12 x = 12 can be expressed as ab, \frac{a}{b}, where a a and b b are coprime positive integers, what is the value of a+b? a + b?.

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