Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 5 \sin{\left(x^{2} \right)} \cos{\left(y^{2} \right)} - 6 \sin{\left(y^{2} \right)} \cos{\left(x^{2} \right)}=44

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 12 x \sin{\left(x^{2} \right)} \sin{\left(y^{2} \right)} + 10 x \cos{\left(x^{2} \right)} \cos{\left(y^{2} \right)} - 10 y y' \sin{\left(x^{2} \right)} \sin{\left(y^{2} \right)} - 12 y y' \cos{\left(x^{2} \right)} \cos{\left(y^{2} \right)} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{x \left(11 \cos{\left(x^{2} - y^{2} \right)} - \cos{\left(x^{2} + y^{2} \right)}\right)}{y \left(11 \cos{\left(x^{2} - y^{2} \right)} + \cos{\left(x^{2} + y^{2} \right)}\right)}