Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 5 \sqrt{2} \sqrt{x} e^{y} - 7 \log{\left(x \right)} \log{\left(y \right)}=49$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 5 \sqrt{2} \sqrt{x} y' e^{y} - \frac{7 y' \log{\left(x \right)}}{y} - \frac{7 \log{\left(y \right)}}{x} - \frac{5 \sqrt{2} e^{y}}{2 \sqrt{x}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{y \left(14 \sqrt{x} \log{\left(y \right)} + 5 \sqrt{2} x e^{y}\right)}{14 x^{\frac{3}{2}} \log{\left(x \right)} + 10 \sqrt{2} x^{2} y e^{y}}$