d-dx-x-x-x-1-3-1-3-1-3-

Question Number 145620 by Study last updated on 06/Jul/21

\frac{d}{d x} (\sqrt[3]{x + \sqrt[3]{x + \sqrt[3]{x + \dots}}}) = ?

Answered by Olaf_Thorendsen last updated on 06/Jul/21

y = f (x) = \sqrt[3]{x + f (x)} = \sqrt[3]{x + y}

y^{3} = x + y

3 y^{2} y^{'} = 1 + y^{'}

y^{'} = \frac{d y}{d x} = \frac{1}{3 y^{2} - 1} = \frac{1}{3 {(x + \sqrt[3]{x + \sqrt[3]{x + \dots}})}^{2 / 3} - 1}