Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 6 \sqrt{x} \cos{\left(y^{2} \right)} - 6 y^{2} e^{x^{2}}=44

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 12 \sqrt{x} y y' \sin{\left(y^{2} \right)} - 12 x y^{2} e^{x^{2}} - 12 y y' e^{x^{2}} + \frac{3 \cos{\left(y^{2} \right)}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- x^{\frac{3}{2}} y^{2} e^{x^{2}} + \frac{\cos{\left(y^{2} \right)}}{4}}{y \left(\sqrt{x} e^{x^{2}} + x \sin{\left(y^{2} \right)}\right)}