Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 2 x^{2} \log{\left(y \right)} + 7 \sin{\left(x^{3} \right)} \cos{\left(y^{2} \right)}=41

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 21 x^{2} \cos{\left(x^{3} \right)} \cos{\left(y^{2} \right)} - \frac{2 x^{2} y'}{y} - 4 x \log{\left(y \right)} - 14 y y' \sin{\left(x^{3} \right)} \sin{\left(y^{2} \right)} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{x y \left(21 x \cos{\left(x^{3} \right)} \cos{\left(y^{2} \right)} - 4 \log{\left(y \right)}\right)}{2 \left(x^{2} + 7 y^{2} \sin{\left(x^{3} \right)} \sin{\left(y^{2} \right)}\right)}