Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 15 \sqrt{x} \sin{\left(y^{2} \right)} - 21 \sqrt{y} \sin{\left(x^{2} \right)}=-41$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 30 \sqrt{x} y y' \cos{\left(y^{2} \right)} - 42 x \sqrt{y} \cos{\left(x^{2} \right)} - \frac{21 y' \sin{\left(x^{2} \right)}}{2 \sqrt{y}} + \frac{15 \sin{\left(y^{2} \right)}}{2 \sqrt{x}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{- 28 x^{\frac{3}{2}} y \cos{\left(x^{2} \right)} + 5 \sqrt{y} \sin{\left(y^{2} \right)}}{7 \sqrt{x} \sin{\left(x^{2} \right)} - 20 x y^{\frac{3}{2}} \cos{\left(y^{2} \right)}}$