Suppose f is differentiable function such that f(g(x)) = x^2 & f'(x)=1+(f(x))^2 then find the value of g'(2)

Question

Suppose f is differentiable function such that  f(g(x)) = x^2 & f'(x)=1+(f(x))^2 then find the value of g'(2)

Mark M. · Accepted Answer

d/dx[f(g(x))] = f'(g(x))g'(x) (by the Chain Rule)

So, since f(g(x)) = x², we have f'(g(x))g'(x) = 2x

When x = 2, we obtain f'(g(2))g'(2) = 4

Since f'(x) = 1 + [f(x)]², f'(g(2)) = 1 + [f(g(2))]² = 1 + 4² = 17

Therefore, 17(g'(2)) = 4

g'(2) = 4/17

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