Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 4 \sqrt{3} \sqrt{x} \sin{\left(y^{2} \right)} - 5 e^{x^{3}} \cos{\left(y^{2} \right)}=8

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 8 \sqrt{3} \sqrt{x} y y' \cos{\left(y^{2} \right)} - 15 x^{2} e^{x^{3}} \cos{\left(y^{2} \right)} + 10 y y' e^{x^{3}} \sin{\left(y^{2} \right)} + \frac{2 \sqrt{3} \sin{\left(y^{2} \right)}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{\frac{15 x^{\frac{5}{2}} e^{x^{3}} \cos{\left(y^{2} \right)}}{2} - \sqrt{3} \sin{\left(y^{2} \right)}}{y \left(5 \sqrt{x} e^{x^{3}} \sin{\left(y^{2} \right)} + 4 \sqrt{3} x \cos{\left(y^{2} \right)}\right)}