Derivative of y=(x^2+7x+1)*e^(x+1)

1. Apply the product rule:

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

   $f{\left(x \right)} = x^{2} + 7 x + 1$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Differentiate $x^{2} + 7 x + 1$ term by term:

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

      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: $7$

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

      The result is: $2 x + 7$

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

   1. Let $u = x + 1$.

   2. The derivative of $e^{u}$ is itself.

   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:

      $$e^{x + 1}$$

   The result is: $\left(2 x + 7\right) e^{x + 1} + \left(x^{2} + 7 x + 1\right) e^{x + 1}$

2. Now simplify:

   $$\left(x^{2} + 9 x + 8\right) e^{x + 1}$$

The answer is:

$$\left(x^{2} + 9 x + 8\right) e^{x + 1}$$