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

$$\frac{d}{dx}\left(\sqrt[3]{x+\sqrt[3]{x+\sqrt[3]{x+...}}}\right)=?$$

Answered by Olaf_Thorendsen last updated on 06/Jul/21

$$y=f\left(x\right)=\sqrt[3]{x+f\left(x\right)}=\sqrt[3]{x+y}$$$$y^3=x+y$$$$3y^2{y}^{\prime }=1+y^{\prime }$$$$y^{\prime }=\frac{dy}{dx}=\frac{1}{3y^2-1}=\frac{1}{3{(x+\sqrt[3]{x+\sqrt[3]{x+...}})}^{2/3}-1}$$

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