Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle x^{2} \log{\left(y \right)} + 10 \sqrt{2} \sqrt{y} \sin{\left(x \right)}=-48$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: \frac{x^{2} y'}{y} + 2 x \log{\left(y \right)} + 10 \sqrt{2} \sqrt{y} \cos{\left(x \right)} + \frac{5 \sqrt{2} y' \sin{\left(x \right)}}{\sqrt{y}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{2 x y^{\frac{3}{2}} \log{\left(y \right)} + 10 \sqrt{2} y^{2} \cos{\left(x \right)}}{x^{2} \sqrt{y} + 5 \sqrt{2} y \sin{\left(x \right)}}$