Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 6 \sqrt{2} x^{3} \sqrt{y} - 8 \log{\left(y \right)} \sin{\left(x \right)}=-29$

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