Derivative of (2-x)^6(5+2x)^4

1. Apply the product rule:

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

   $f{\left(x \right)} = \left(2 - x\right)^{6}$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Let $u = 2 - x$.

   2. Apply the power rule: $u^{6}$ goes to $6 u^{5}$

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 - x\right)$:

      1. Differentiate $2 - x$ term by term:

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

         2. 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: $-1$

         The result is: $-1$

      The result of the chain rule is:

      $$- 6 \left(2 - x\right)^{5}$$

   $g{\left(x \right)} = \left(2 x + 5\right)^{4}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = 2 x + 5$.

   2. Apply the power rule: $u^{4}$ goes to $4 u^{3}$

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x + 5\right)$:

      1. Differentiate $2 x + 5$ 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. Apply the power rule: $x$ goes to $1$

            So, the result is: $2$

         The result is: $2$

      The result of the chain rule is:

      $$8 \left(2 x + 5\right)^{3}$$

   The result is: $8 \left(2 - x\right)^{6} \left(2 x + 5\right)^{3} - 6 \left(2 - x\right)^{5} \left(2 x + 5\right)^{4}$

2. Now simplify:

   $$\left(x - 2\right)^{5} \left(2 x + 5\right)^{3} \left(20 x + 14\right)$$

The answer is:

$$\left(x - 2\right)^{5} \left(2 x + 5\right)^{3} \left(20 x + 14\right)$$