Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 7 \sqrt{7} \sqrt{x} \sin{\left(y^{2} \right)} - \log{\left(x \right)} \log{\left(y \right)}=44$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 14 \sqrt{7} \sqrt{x} y y' \cos{\left(y^{2} \right)} - \frac{y' \log{\left(x \right)}}{y} - \frac{\log{\left(y \right)}}{x} - \frac{7 \sqrt{7} \sin{\left(y^{2} \right)}}{2 \sqrt{x}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{y \left(2 \sqrt{x} \log{\left(y \right)} + 7 \sqrt{7} x \sin{\left(y^{2} \right)}\right)}{2 x^{\frac{3}{2}} \log{\left(x \right)} + 28 \sqrt{7} x^{2} y^{2} \cos{\left(y^{2} \right)}}$