Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 6 \log{\left(y \right)} \cos{\left(x \right)} + 8 \sin{\left(x^{2} \right)} \sin{\left(y^{2} \right)}=47$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 16 x \sin{\left(y^{2} \right)} \cos{\left(x^{2} \right)} + 16 y y' \sin{\left(x^{2} \right)} \cos{\left(y^{2} \right)} + 6 \log{\left(y \right)} \sin{\left(x \right)} - \frac{6 y' \cos{\left(x \right)}}{y} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{y \left(8 x \sin{\left(y^{2} \right)} \cos{\left(x^{2} \right)} + 3 \log{\left(y \right)} \sin{\left(x \right)}\right)}{- 8 y^{2} \sin{\left(x^{2} \right)} \cos{\left(y^{2} \right)} + 3 \cos{\left(x \right)}}$