Derivative of Inverse Function at A Given Point

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The formula for the derivative of an inverse function is nothing more than a peculiar application of the chain rule. Suppose f and g are inverse functions of one another. Then f(g(x)) = x. It follows that d[f(g(x))]/dx = f'(g(x))·g'(x) = dx/dx = 1. Hence g'(x) = 1/f'(g(x)). Considering your example, where we know f(x) = x^2 but we don't know that g(x) = sqrt[x], or if we do know that g(x) is sqrt[x], we don't know how to compute its derivative, we can apply the formula in the following way: We want g'(9). f'(x) = 2x => f'(g(x)) = 2g(x) => f'(g(9)) = 2g(9). Now, f(3) = 9 => g(f(3)) = 3 = g(9). Thus f'(g(9)) = 2·3 = 6. Finally, g'(9) = 1/f'(g(9)) = 1/6 ◼

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