Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 12 \sqrt{2} \sqrt{y} \log{\left(x \right)} + \sin{\left(x^{2} \right)} \cos{\left(y \right)}=-41

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 2 x \cos{\left(x^{2} \right)} \cos{\left(y \right)} - y' \sin{\left(x^{2} \right)} \sin{\left(y \right)} + \frac{6 \sqrt{2} y' \log{\left(x \right)}}{\sqrt{y}} + \frac{12 \sqrt{2} \sqrt{y}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{2 \left(x^{2} \sqrt{y} \cos{\left(x^{2} \right)} \cos{\left(y \right)} + 6 \sqrt{2} y\right)}{x \left(\sqrt{y} \sin{\left(x^{2} \right)} \sin{\left(y \right)} - 6 \sqrt{2} \log{\left(x \right)}\right)}