Derivative of 5xe^(5x)*x+e^(5x)

1. Differentiate $5 x x e^{5 x} + e^{5 x}$ term by term:

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

      1. Apply the product rule:

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

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

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

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

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

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

         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: $5 x^{2} e^{5 x} + 2 x e^{5 x}$

      So, the result is: $25 x^{2} e^{5 x} + 10 x e^{5 x}$

   2. Let $u = 5 x$.

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

   4. 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: $25 x^{2} e^{5 x} + 10 x e^{5 x} + 5 e^{5 x}$

2. Now simplify:

   $$\left(25 x^{2} + 10 x + 5\right) e^{5 x}$$

The answer is:

$$\left(25 x^{2} + 10 x + 5\right) e^{5 x}$$