Derivative of y=(x+2)²(2x-1)³

1. Apply the product rule:

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

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

   1. Let $u = x + 2$.

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

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

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

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

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

         The result is: $1$

      The result of the chain rule is:

      $$2 x + 4$$

   $g{\left(x \right)} = \left(2 x - 1\right)^{3}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = 2 x - 1$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x - 1\right)$:

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

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

         The result is: $2$

      The result of the chain rule is:

      $$6 \left(2 x - 1\right)^{2}$$

   The result is: $6 \left(x + 2\right)^{2} \left(2 x - 1\right)^{2} + \left(2 x - 1\right)^{3} \left(2 x + 4\right)$

2. Now simplify:

   $$10 \left(x + 1\right) \left(x + 2\right) \left(2 x - 1\right)^{2}$$

The answer is:

$$10 \left(x + 1\right) \left(x + 2\right) \left(2 x - 1\right)^{2}$$