Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 3 \sqrt{5} \sqrt{y} e^{x} - \cos{\left(x^{3} \right)} \cos{\left(y^{3} \right)}=-44$

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