Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 8 \sqrt{x} \sin{\left(y^{2} \right)} + 3 \log{\left(y \right)} \sin{\left(x^{2} \right)}=14$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 16 \sqrt{x} y y' \cos{\left(y^{2} \right)} + 6 x \log{\left(y \right)} \cos{\left(x^{2} \right)} + \frac{3 y' \sin{\left(x^{2} \right)}}{y} + \frac{4 \sin{\left(y^{2} \right)}}{\sqrt{x}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{2 y \left(3 x^{\frac{3}{2}} \log{\left(y \right)} \cos{\left(x^{2} \right)} + 2 \sin{\left(y^{2} \right)}\right)}{3 \sqrt{x} \sin{\left(x^{2} \right)} + 16 x y^{2} \cos{\left(y^{2} \right)}}$