Derivative of 1/4(x+1)^2

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

   1. Let $u = x + 1$.

   2. Apply the power rule: $u^{2}$ goes to $2 u$

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

      1. Differentiate $x + 1$ term by term:

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

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

         The result is: $1$

      The result of the chain rule is:

      $$2 x + 2$$

   So, the result is: $\frac{x}{2} + \frac{1}{2}$

The answer is:

$$\frac{x}{2} + \frac{1}{2}$$