Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle e^{x^{2}} \log{\left(y \right)} + 6 \sin{\left(y^{2} \right)} \cos{\left(x \right)}=23$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 2 x e^{x^{2}} \log{\left(y \right)} + 12 y y' \cos{\left(x \right)} \cos{\left(y^{2} \right)} - 6 \sin{\left(x \right)} \sin{\left(y^{2} \right)} + \frac{y' e^{x^{2}}}{y} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{2 y \left(- x e^{x^{2}} \log{\left(y \right)} + 3 \sin{\left(x \right)} \sin{\left(y^{2} \right)}\right)}{12 y^{2} \cos{\left(x \right)} \cos{\left(y^{2} \right)} + e^{x^{2}}}$