Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 7 \sqrt{7} x^{2} \sqrt{y} - 4 \log{\left(x \right)} \sin{\left(y^{2} \right)}=-49

Taking the derivative of both sides using implicit differentiation gives: LaTeX: \frac{7 \sqrt{7} x^{2} y'}{2 \sqrt{y}} + 14 \sqrt{7} x \sqrt{y} - 8 y y' \log{\left(x \right)} \cos{\left(y^{2} \right)} - \frac{4 \sin{\left(y^{2} \right)}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{4 \left(- 7 \sqrt{7} x^{2} y + 2 \sqrt{y} \sin{\left(y^{2} \right)}\right)}{x \left(7 \sqrt{7} x^{2} - 16 y^{\frac{3}{2}} \log{\left(x \right)} \cos{\left(y^{2} \right)}\right)}