Derivative of x*exp(3x-7)

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 $1$

   $g{\left(x \right)} = e^{3 x - 7}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = 3 x - 7$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(3 x - 7\right)$:

      1. Differentiate $3 x - 7$ 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: $3$

         2. The derivative of the constant $-7$ is zero.

         The result is: $3$

      The result of the chain rule is:

      $$3 e^{3 x - 7}$$

   The result is: $3 x e^{3 x - 7} + e^{3 x - 7}$

2. Now simplify:

   $$\left(3 x + 1\right) e^{3 x - 7}$$

The answer is:

$$\left(3 x + 1\right) e^{3 x - 7}$$