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Let f(x) = x³ + 3x + 4 and let g be the inverse function of f. Find the value of d/dx [g(x) / g(g(x))] at x = 4.

1. -1/3
2. -1/2
3. 3
4. 6

Correct answer: -1/3

Solution

f(x) = x³ + 3x + 4. f(0) = 4, so g(4) = 0. f(-1) = -1 - 3 + 4 = 0, so g(0) = -1. g(g(4)) = g(0) = -1. f'(x) = 3x² + 3. g'(x) = 1/f'(g(x)). g'(4) = 1/f'(0) = 1/3. g'(0) = 1/f'(-1) = 1/(3*1+3) = 1/6. Let h(x) = g(x)/g(g(x)). At x=4: g(4)=0, g(g(4))=-1. h(4) = 0/(-1) = 0. h'(x) = [g'(x)*g(g(x)) - g(x)*g'(g(x))*g'(x)] / [g(g(x))]². At x=4: h'(4) = [g'(4)*(-1) - 0*g'(0)*g'(4)] / (-1)² = [(1/3)*(-1) - 0] / 1 = -1/3.

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