Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle x^{2} \sin{\left(y^{2} \right)} + 5 e^{x} \log{\left(y \right)}=22$

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