Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 3 \sqrt{y} \cos{\left(x^{2} \right)} + 2 e^{x^{2}} \log{\left(y \right)}=25

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 6 x \sqrt{y} \sin{\left(x^{2} \right)} + 4 x e^{x^{2}} \log{\left(y \right)} + \frac{2 y' e^{x^{2}}}{y} + \frac{3 y' \cos{\left(x^{2} \right)}}{2 \sqrt{y}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{4 x \left(- 2 y^{\frac{3}{2}} e^{x^{2}} \log{\left(y \right)} + 3 y^{2} \sin{\left(x^{2} \right)}\right)}{4 \sqrt{y} e^{x^{2}} + 3 y \cos{\left(x^{2} \right)}}