Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 9 \sqrt{2} \sqrt{y} \cos{\left(x^{2} \right)} + 4 \sin{\left(x^{2} \right)} \sin{\left(y^{2} \right)}=38$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 18 \sqrt{2} x \sqrt{y} \sin{\left(x^{2} \right)} + 8 x \sin{\left(y^{2} \right)} \cos{\left(x^{2} \right)} + 8 y y' \sin{\left(x^{2} \right)} \cos{\left(y^{2} \right)} + \frac{9 \sqrt{2} y' \cos{\left(x^{2} \right)}}{2 \sqrt{y}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{4 x \left(- 4 \sqrt{y} \sin{\left(y^{2} \right)} \cos{\left(x^{2} \right)} + 9 \sqrt{2} y \sin{\left(x^{2} \right)}\right)}{16 y^{\frac{3}{2}} \sin{\left(x^{2} \right)} \cos{\left(y^{2} \right)} + 9 \sqrt{2} \cos{\left(x^{2} \right)}}$