Derivative of z=ln(1+2^x)

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

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

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2^{x} + 1\right)$:

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

      1. The derivative of the constant $1$ is zero.

      2. $\frac{d}{d x} 2^{x} = 2^{x} \log{\left(2 \right)}$

      The result is: $2^{x} \log{\left(2 \right)}$

   The result of the chain rule is:

   $$\frac{2^{x} \log{\left(2 \right)}}{2^{x} + 1}$$

The answer is:

$$\frac{2^{x} \log{\left(2 \right)}}{2^{x} + 1}$$