Answer
$\dfrac {3\cos \left( \tan \sqrt {1+x^{3}}\right) \times x^{2}}{2\sqrt {1+x^{3}}\times \cos \sqrt {1+x^{3}}}
$
Work Step by Step
$\dfrac {d}{dx}\sin \left( \tan \sqrt {1+x^{3}}\right) =\cos \left( \tan \sqrt {1+x^{3}}\right) \times \left( \dfrac {d}{dx}\tan \sqrt {1+x^{3}}\right) =\dfrac {\cos \left( \tan \sqrt {1+x^{3}}\right) }{\cos^ 2\sqrt {1+x^{3}}}\times \left( \dfrac {d}{dx}\sqrt {1+x^{3}}\right) =\dfrac {\cos \left( \tan \sqrt {1+x^{3}}\right) }{\cos \sqrt {1+x^{3}}}\times \dfrac {1}{2\sqrt {1+x^{3}}}\times \dfrac {d}{dx}\left( 1+x^{3}\right) =\dfrac {3\cos \left( \tan \sqrt {1+x^{3}}\right) \times x^{2}}{2\sqrt {1+x^{3}}\times \cos \sqrt {1+x^{3}}}
$