Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 6 \sqrt{6} \sqrt{x} \log{\left(y \right)} + 7 \sin{\left(x^{2} \right)} \sin{\left(y^{2} \right)}=38

Taking the derivative of both sides using implicit differentiation gives: LaTeX: \frac{6 \sqrt{6} \sqrt{x} y'}{y} + 14 x \sin{\left(y^{2} \right)} \cos{\left(x^{2} \right)} + 14 y y' \sin{\left(x^{2} \right)} \cos{\left(y^{2} \right)} + \frac{3 \sqrt{6} \log{\left(y \right)}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = - \frac{y \left(14 x^{\frac{3}{2}} \sin{\left(y^{2} \right)} \cos{\left(x^{2} \right)} + 3 \sqrt{6} \log{\left(y \right)}\right)}{14 \sqrt{x} y^{2} \sin{\left(x^{2} \right)} \cos{\left(y^{2} \right)} + 6 \sqrt{6} x}