Derivative of (5-6*x)*exp(8*x)

1. Apply the product rule:

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

   $f{\left(x \right)} = 5 - 6 x$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Differentiate $5 - 6 x$ term by term:

      1. The derivative of the constant $5$ is zero.

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

         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: $6$

         So, the result is: $-6$

      The result is: $-6$

   $g{\left(x \right)} = e^{8 x}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = 8 x$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 8 x$:

      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: $8$

      The result of the chain rule is:

      $$8 e^{8 x}$$

   The result is: $8 \cdot \left(5 - 6 x\right) e^{8 x} - 6 e^{8 x}$

2. Now simplify:

   $$\left(34 - 48 x\right) e^{8 x}$$

The answer is:

$$\left(34 - 48 x\right) e^{8 x}$$