Derivative of exp(5*x)+x*exp(5*x)+1/2*(x*x*exp(5*x))

1. Differentiate $\frac{x x e^{5 x}}{2} + \left(x e^{5 x} + e^{5 x}\right)$ term by term:

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

      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:

         $$5 e^{5 x}$$

      4. Apply the product rule:

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

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

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

         $g{\left(x \right)} = e^{5 x}$; 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: $5 x e^{5 x} + e^{5 x}$

      The result is: $5 x e^{5 x} + 6 e^{5 x}$

   2. 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: $\frac{5 x^{2} e^{5 x}}{2} + x e^{5 x}$

   The result is: $\frac{5 x^{2} e^{5 x}}{2} + 6 x e^{5 x} + 6 e^{5 x}$

2. Now simplify:

   $$\frac{\left(5 x^{2} + 12 x + 12\right) e^{5 x}}{2}$$

The answer is:

$$\frac{\left(5 x^{2} + 12 x + 12\right) e^{5 x}}{2}$$