Derivative of y=ln(e^x+2x)

1. Let $u = 2 x + e^{x}$.

2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$.

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x + e^{x}\right)$:

   1. Differentiate $2 x + e^{x}$ term by term:

      1. The derivative of $e^{x}$ is itself.

      2. The derivative of a constant times a function is the constant times the derivative of the function.

         1. Apply the power rule: $x$ goes to $1$

         So, the result is: $2$

      The result is: $e^{x} + 2$

   The result of the chain rule is:

   $$\frac{e^{x} + 2}{2 x + e^{x}}$$

4. Now simplify:

   $$\frac{e^{x} + 2}{2 x + e^{x}}$$

The answer is: