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

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