Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 5 \log{\left(x \right)} \sin{\left(y^{2} \right)} - \log{\left(y \right)} \sin{\left(x^{2} \right)}=-35

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 2 x \log{\left(y \right)} \cos{\left(x^{2} \right)} - 10 y y' \log{\left(x \right)} \cos{\left(y^{2} \right)} - \frac{y' \sin{\left(x^{2} \right)}}{y} - \frac{5 \sin{\left(y^{2} \right)}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = - \frac{y \left(2 x^{2} \log{\left(y \right)} \cos{\left(x^{2} \right)} + 5 \sin{\left(y^{2} \right)}\right)}{x \left(10 y^{2} \log{\left(x \right)} \cos{\left(y^{2} \right)} + \sin{\left(x^{2} \right)}\right)}