Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 12 \sqrt{x} \sin{\left(y^{2} \right)} - 7 e^{x} \log{\left(y \right)}=-42$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 24 \sqrt{x} y y' \cos{\left(y^{2} \right)} - 7 e^{x} \log{\left(y \right)} - \frac{7 y' e^{x}}{y} + \frac{6 \sin{\left(y^{2} \right)}}{\sqrt{x}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{y \left(- 7 \sqrt{x} e^{x} \log{\left(y \right)} + 6 \sin{\left(y^{2} \right)}\right)}{7 \sqrt{x} e^{x} - 24 x y^{2} \cos{\left(y^{2} \right)}}$