Derivative of (x+4)^2*(x+1)+9

1. Differentiate $\left(x + 1\right) \left(x + 4\right)^{2} + 9$ term by term:

   1. Apply the product rule:

      $$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$$

      $f{\left(x \right)} = \left(x + 4\right)^{2}$; to find $\frac{d}{d x} f{\left(x \right)}$:

      1. Let $u = x + 4$.

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

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 4\right)$:

         1. Differentiate $x + 4$ term by term:

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

            2. The derivative of the constant $4$ is zero.

            The result is: $1$

         The result of the chain rule is:

         $$2 x + 8$$

      $g{\left(x \right)} = x + 1$; to find $\frac{d}{d x} g{\left(x \right)}$:

      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 is: $\left(x + 1\right) \left(2 x + 8\right) + \left(x + 4\right)^{2}$

   2. The derivative of the constant $9$ is zero.

   The result is: $\left(x + 1\right) \left(2 x + 8\right) + \left(x + 4\right)^{2}$

2. Now simplify:

   $$3 \left(x + 2\right) \left(x + 4\right)$$

The answer is:

$$3 \left(x + 2\right) \left(x + 4\right)$$