Derivative of (x^2+1)e^(5x)

1. Apply the product rule:

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

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

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

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

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

      The result is: $2 x$

   $g{\left(x \right)} = e^{5 x}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = 5 x$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 5 x$:

      1. 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: $5$

      The result of the chain rule is:

      $$5 e^{5 x}$$

   The result is: $2 x e^{5 x} + 5 \left(x^{2} + 1\right) e^{5 x}$

2. Now simplify:

   $$\left(5 x^{2} + 2 x + 5\right) e^{5 x}$$

The answer is:

$$\left(5 x^{2} + 2 x + 5\right) e^{5 x}$$