Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 16 \sqrt{x} \sqrt{y} - 7 \sin{\left(y^{2} \right)} \cos{\left(x^{2} \right)}=-49

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