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)} + 5 x^{2} \cos{\left(y^{3} \right)}=45

Taking the derivative of both sides using implicit differentiation gives: LaTeX: \frac{6 \sqrt{x} y'}{y} - 15 x^{2} y^{2} y' \sin{\left(y^{3} \right)} + 10 x \cos{\left(y^{3} \right)} + \frac{3 \log{\left(y \right)}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{y \left(10 x^{\frac{3}{2}} \cos{\left(y^{3} \right)} + 3 \log{\left(y \right)}\right)}{3 \left(5 x^{\frac{5}{2}} y^{3} \sin{\left(y^{3} \right)} - 2 x\right)}