Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 5 x^{3} \sqrt{y} - 9 \log{\left(x \right)} \sin{\left(y \right)}=48

Taking the derivative of both sides using implicit differentiation gives: LaTeX: \frac{5 x^{3} y'}{2 \sqrt{y}} + 15 x^{2} \sqrt{y} - 9 y' \log{\left(x \right)} \cos{\left(y \right)} - \frac{9 \sin{\left(y \right)}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{6 \left(- 5 x^{3} y + 3 \sqrt{y} \sin{\left(y \right)}\right)}{x \left(5 x^{3} - 18 \sqrt{y} \log{\left(x \right)} \cos{\left(y \right)}\right)}