Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 5 \sqrt{7} \sqrt{y} e^{x^{2}} + 3 e^{y^{2}} \sin{\left(x^{3} \right)}=30$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 9 x^{2} e^{y^{2}} \cos{\left(x^{3} \right)} + 10 \sqrt{7} x \sqrt{y} e^{x^{2}} + 6 y y' e^{y^{2}} \sin{\left(x^{3} \right)} + \frac{5 \sqrt{7} y' e^{x^{2}}}{2 \sqrt{y}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{2 x \left(9 x \sqrt{y} e^{y^{2}} \cos{\left(x^{3} \right)} + 10 \sqrt{7} y e^{x^{2}}\right)}{12 y^{\frac{3}{2}} e^{y^{2}} \sin{\left(x^{3} \right)} + 5 \sqrt{7} e^{x^{2}}}$