Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 7 \sqrt{3} \sqrt{x} \cos{\left(y^{2} \right)} - 2 e^{y} \log{\left(x \right)}=-40$

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