Answer
$$y' = \frac{{200y\sqrt {3{x^7} + {y^2}} - 21{x^6}}}{{y - 2\sin 2y\sqrt {3{x^7} + {y^2}} - 200\sqrt {3{x^7} + {y^2}} x}}$$
Work Step by Step
$$\eqalign{
& \sqrt {3{x^7} + {y^2}} = {\sin ^2}y + 100xy \cr
& {\text{Differentiate both sides with respect to }}x \cr
& \frac{{21{x^6} + 2yy'}}{{2\sqrt {3{x^7} + {y^2}} }} = \left( {2\sin y\cos y} \right)y' + 100xy' + 100y \cr
& \frac{{21{x^6}}}{{2\sqrt {3{x^7} + {y^2}} }} + \frac{{yy'}}{{2\sqrt {3{x^7} + {y^2}} }} = y'\sin 2y + 100xy' + 100y \cr
& {\text{}} \cr
& \frac{{yy'}}{{2\sqrt {3{x^7} + {y^2}} }} - y'\sin 2y - 100xy' = 100y - \frac{{21{x^6}}}{{2\sqrt {3{x^7} + {y^2}} }} \cr
& \left( {\frac{y}{{2\sqrt {3{x^7} + {y^2}} }} - \sin 2y - 100x} \right)y' = 100y - \frac{{21{x^6}}}{{2\sqrt {3{x^7} + {y^2}} }} \cr
& {\text{Solve for }}y' \cr
& y' = \frac{{100y - \frac{{21{x^6}}}{{2\sqrt {3{x^7} + {y^2}} }}}}{{\frac{y}{{2\sqrt {3{x^7} + {y^2}} }} - \sin 2y - 100x}} \cr
& y' = \frac{{\frac{{200y\sqrt {3{x^7} + {y^2}} - 21{x^6}}}{{2\sqrt {3{x^7} + {y^2}} }}}}{{\frac{{y - 2\sin 2y\sqrt {3{x^7} + {y^2}} - 200\sqrt {3{x^7} + {y^2}} x}}{{2\sqrt {3{x^7} + {y^2}} }}}} \cr
& y' = \frac{{200y\sqrt {3{x^7} + {y^2}} - 21{x^6}}}{{y - 2\sin 2y\sqrt {3{x^7} + {y^2}} - 200\sqrt {3{x^7} + {y^2}} x}} \cr} $$
