Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle e^{y^{3}} \sin{\left(x \right)} + 3 \log{\left(x \right)} \sin{\left(y \right)}=22

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