Derivative of (x-1)^4(x+1)^7

1. Apply the product rule:

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

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

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

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

         The result is: $1$

      The result of the chain rule is:

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

   $g{\left(x \right)} = \left(x + 1\right)^{7}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = x + 1$.

   2. Apply the power rule: $u^{7}$ goes to $7 u^{6}$

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

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

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

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

         The result is: $1$

      The result of the chain rule is:

      $$7 \left(x + 1\right)^{6}$$

   The result is: $7 \left(x - 1\right)^{4} \left(x + 1\right)^{6} + 4 \left(x - 1\right)^{3} \left(x + 1\right)^{7}$

2. Now simplify:

   $$\left(x - 1\right)^{3} \left(x + 1\right)^{6} \left(11 x - 3\right)$$

The answer is:

$$\left(x - 1\right)^{3} \left(x + 1\right)^{6} \left(11 x - 3\right)$$