Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 8 \sqrt{5} \sqrt{x} \sin{\left(y \right)} + 7 e^{y^{2}} \log{\left(x \right)}=27

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 8 \sqrt{5} \sqrt{x} y' \cos{\left(y \right)} + 14 y y' e^{y^{2}} \log{\left(x \right)} + \frac{7 e^{y^{2}}}{x} - \frac{4 \sqrt{5} \sin{\left(y \right)}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- 7 \sqrt{x} e^{y^{2}} + 4 \sqrt{5} x \sin{\left(y \right)}}{2 \left(7 x^{\frac{3}{2}} y e^{y^{2}} \log{\left(x \right)} - 4 \sqrt{5} x^{2} \cos{\left(y \right)}\right)}