Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 3 x \sin{\left(y \right)} - 2 \sin{\left(x \right)} \cos{\left(y^{2} \right)}=42$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 3 x y' \cos{\left(y \right)} + 4 y y' \sin{\left(x \right)} \sin{\left(y^{2} \right)} - 3 \sin{\left(y \right)} - 2 \cos{\left(x \right)} \cos{\left(y^{2} \right)} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{3 \sin{\left(y \right)} + 2 \cos{\left(x \right)} \cos{\left(y^{2} \right)}}{3 x \cos{\left(y \right)} - 4 y \sin{\left(x \right)} \sin{\left(y^{2} \right)}}$