Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 8 \sqrt{x} e^{y} - 2 \log{\left(y \right)} \cos{\left(x^{2} \right)}=35

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