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

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