Derivative of (xe^(5x))(ax+b)

1. Apply the product rule:

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

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

   1. Apply the product rule:

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

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

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

   1. Differentiate $a x + b$ term by term:

      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: $a$

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

      The result is: $a$

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

2. Now simplify:

   $$\left(a x + \left(5 x + 1\right) \left(a x + b\right)\right) e^{5 x}$$

The answer is:

$$\left(a x + \left(5 x + 1\right) \left(a x + b\right)\right) e^{5 x}$$