Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 3 \sqrt{x} \log{\left(y \right)} - 5 \log{\left(x \right)} \cos{\left(y^{2} \right)}=6

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