Derivative of (2*x-1)^5*(1+x)^4

1. Apply the product rule:

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

   $f{\left(x \right)} = \left(2 x - 1\right)^{5}$; to find $\frac{d}{d x} f{\left(x \right)}$:

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

   2. Apply the power rule: $u^{5}$ goes to $5 u^{4}$

   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:

      $$10 \left(2 x - 1\right)^{4}$$

   $g{\left(x \right)} = \left(x + 1\right)^{4}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = x + 1$.

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

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

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

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

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

         The result is: $1$

      The result of the chain rule is:

      $$4 \left(x + 1\right)^{3}$$

   The result is: $10 \left(x + 1\right)^{4} \left(2 x - 1\right)^{4} + 4 \left(x + 1\right)^{3} \left(2 x - 1\right)^{5}$

2. Now simplify:

   $$\left(x + 1\right)^{3} \left(2 x - 1\right)^{4} \left(18 x + 6\right)$$

The answer is:

$$\left(x + 1\right)^{3} \left(2 x - 1\right)^{4} \left(18 x + 6\right)$$