Answer: [tex]\dfrac{x^2}{(x^3+1)}[/tex]
Step-by-step explanation
Properties of derivative , we use here :
The given function : [tex]y=\ln(x^3+1)^{\frac{1}{3}}[/tex]
[tex]y=\dfrac{1}{3}\cdot \ln (x^3+1)[/tex] [[tex]\because n\ln x=\ln x^n[/tex]]
Now , Differentiate both sides , we get
[tex]\dfrac{dy}{dx}=\dfrac{1}{3}\cdot \dfrac{1}{(x^3+1)}\cdot \dfrac{d}{dx}(x^3+1)[/tex]
(By chain rule)
[tex]=\dfrac{1}{3}\cdot \dfrac{1}{(x^3+1)}\cdot (3x^2+0)[/tex]
[tex]=\dfrac{1}{3}\cdot \dfrac{1}{(x^3+1)}(3x^2)[/tex]
[tex]=\dfrac{x^2}{(x^3+1)}[/tex]
Hence, the derivative of the function will be : [tex]\dfrac{x^2}{(x^3+1)}[/tex]
