Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 6 \sqrt{x} \log{\left(y \right)} + 3 y^{3} \cos{\left(x \right)}=44

Taking the derivative of both sides using implicit differentiation gives: LaTeX: \frac{6 \sqrt{x} y'}{y} - 3 y^{3} \sin{\left(x \right)} + 9 y^{2} y' \cos{\left(x \right)} + \frac{3 \log{\left(y \right)}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{y \left(\sqrt{x} y^{3} \sin{\left(x \right)} - \log{\left(y \right)}\right)}{3 \sqrt{x} y^{3} \cos{\left(x \right)} + 2 x}