Derivative of y=e^x*(x^30+30x+1)

1. Apply the product rule:

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

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

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

   $g{\left(x \right)} = x^{30} + 30 x + 1$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $x^{30} + 30 x + 1$ term by term:

      1. Apply the power rule: $x^{30}$ goes to $30 x^{29}$

      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: $30$

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

      The result is: $30 x^{29} + 30$

   The result is: $\left(30 x^{29} + 30\right) e^{x} + \left(x^{30} + 30 x + 1\right) e^{x}$

2. Now simplify:

   $$\left(x^{30} + 30 x^{29} + 30 x + 31\right) e^{x}$$

The answer is:

$$\left(x^{30} + 30 x^{29} + 30 x + 31\right) e^{x}$$