Consider the given function :
[tex]{:\implies \quad \sf y=x\sqrt[3]{x}}[/tex]
Before doing this question let's recall some basic formulae of calculus;
- [tex]{\boxed{\bf{a^{m}\cdot{a^{n}}=a^{m+n}}}}[/tex]
- [tex]{\boxed{\bf{\dfrac{d}{dx}(x^n)=n{x}^{n-1}}}}[/tex]
So, now let's move on to our question ;
[tex]{:\implies \quad \sf y=x\sqrt[3]{x}}[/tex]
Can be further written as ;
[tex]{:\implies \quad \sf y=x(x)^{\footnotesize \dfrac13}}[/tex]
Using the 1st identity can be written as ;
[tex]{:\implies \quad \sf y=x^{\footnotesize \dfrac43}}[/tex]
Now, differentiating both sides w.r.t.x ;
[tex]{:\implies \quad \sf \dfrac{d}{dx}(y)=\dfrac{4}{3}(x)^{\footnotesize \dfrac{4}{3}-1}}[/tex]
[tex]{:\implies \quad \sf \dfrac{dy}{dx}=\dfrac{4}{3}(x)^{\footnotesize \dfrac{4-3}{3}}}[/tex]
[tex]{:\implies \quad \sf \dfrac{dy}{dx}=\dfrac{4}{3}(x)^{\footnotesize \dfrac{1}{3}}}[/tex]
[tex]{:\implies \quad \bf \therefore \quad \underline{\underline{\dfrac{dy}{dx}=\dfrac{4}{3}\sqrt[3]{x}}}}[/tex]
Hence, Option B) is correct
