Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 7 \sqrt{3} \sqrt{x} \sin{\left(y^{2} \right)} - 9 x^{2} \cos{\left(y^{2} \right)}=-41

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 14 \sqrt{3} \sqrt{x} y y' \cos{\left(y^{2} \right)} + 18 x^{2} y y' \sin{\left(y^{2} \right)} - 18 x \cos{\left(y^{2} \right)} + \frac{7 \sqrt{3} \sin{\left(y^{2} \right)}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{36 x^{\frac{3}{2}} \cos{\left(y^{2} \right)} - 7 \sqrt{3} \sin{\left(y^{2} \right)}}{4 y \left(9 x^{\frac{5}{2}} \sin{\left(y^{2} \right)} + 7 \sqrt{3} x \cos{\left(y^{2} \right)}\right)}