Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 3 \sqrt{5} x^{3} \sqrt{y} + 5 e^{y^{2}} \cos{\left(x^{2} \right)}=23$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - \frac{3 \sqrt{5} x^{3} y'}{2 \sqrt{y}} - 9 \sqrt{5} x^{2} \sqrt{y} - 10 x e^{y^{2}} \sin{\left(x^{2} \right)} + 10 y y' e^{y^{2}} \cos{\left(x^{2} \right)} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{2 x \left(9 \sqrt{5} x y + 10 \sqrt{y} e^{y^{2}} \sin{\left(x^{2} \right)}\right)}{3 \sqrt{5} x^{3} - 20 y^{\frac{3}{2}} e^{y^{2}} \cos{\left(x^{2} \right)}}$