Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 18 \sqrt{y} \sin{\left(x^{2} \right)} + y^{3} \log{\left(x \right)}=9

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 36 x \sqrt{y} \cos{\left(x^{2} \right)} + 3 y^{2} y' \log{\left(x \right)} - \frac{9 y' \sin{\left(x^{2} \right)}}{\sqrt{y}} + \frac{y^{3}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{36 x^{2} y \cos{\left(x^{2} \right)} - y^{\frac{7}{2}}}{3 x \left(y^{\frac{5}{2}} \log{\left(x \right)} - 3 \sin{\left(x^{2} \right)}\right)}