Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 9 \sqrt{7} \sqrt{x} y + 7 \log{\left(x \right)} \cos{\left(y^{2} \right)}=12

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 9 \sqrt{7} \sqrt{x} y' - 14 y y' \log{\left(x \right)} \sin{\left(y^{2} \right)} + \frac{7 \cos{\left(y^{2} \right)}}{x} + \frac{9 \sqrt{7} y}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{14 \sqrt{x} \cos{\left(y^{2} \right)} + 9 \sqrt{7} x y}{2 \left(14 x^{\frac{3}{2}} y \log{\left(x \right)} \sin{\left(y^{2} \right)} - 9 \sqrt{7} x^{2}\right)}