Derivative of xe^(-5x)+4

1. Differentiate $e^{- 5 x} x + 4$ term by term:

   1. Apply the quotient rule, which is:

      $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

      $f{\left(x \right)} = x$ and $g{\left(x \right)} = e^{5 x}$.

      To find $\frac{d}{d x} f{\left(x \right)}$:

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

      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:

         $$5 e^{5 x}$$

      Now plug in to the quotient rule:

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

   2. The derivative of the constant $4$ is zero.

   The result is: $\left(- 5 x e^{5 x} + e^{5 x}\right) e^{- 10 x}$

2. Now simplify:

   $$\left(1 - 5 x\right) e^{- 5 x}$$

The answer is:

$$\left(1 - 5 x\right) e^{- 5 x}$$