Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 6 \sqrt{y} \sin{\left(x \right)} - 5 e^{x^{2}} \log{\left(y \right)}=-7$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 10 x e^{x^{2}} \log{\left(y \right)} - 6 \sqrt{y} \cos{\left(x \right)} - \frac{5 y' e^{x^{2}}}{y} - \frac{3 y' \sin{\left(x \right)}}{\sqrt{y}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{10 x y^{\frac{3}{2}} e^{x^{2}} \log{\left(y \right)} + 6 y^{2} \cos{\left(x \right)}}{5 \sqrt{y} e^{x^{2}} + 3 y \sin{\left(x \right)}}$