Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 5 \sqrt{7} \sqrt{x} \log{\left(y \right)} + \sin{\left(x^{2} \right)} \sin{\left(y^{2} \right)}=18

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - \frac{5 \sqrt{7} \sqrt{x} y'}{y} + 2 x \sin{\left(y^{2} \right)} \cos{\left(x^{2} \right)} + 2 y y' \sin{\left(x^{2} \right)} \cos{\left(y^{2} \right)} - \frac{5 \sqrt{7} \log{\left(y \right)}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{y \left(- 4 x^{\frac{3}{2}} \sin{\left(y^{2} \right)} \cos{\left(x^{2} \right)} + 5 \sqrt{7} \log{\left(y \right)}\right)}{2 \left(2 \sqrt{x} y^{2} \sin{\left(x^{2} \right)} \cos{\left(y^{2} \right)} - 5 \sqrt{7} x\right)}