Derivative of (x+2)^5(x-3)^4

1. Apply the product rule:

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

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

   1. Let $u = x + 2$.

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

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

      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:

      $$5 \left(x + 2\right)^{4}$$

   $g{\left(x \right)} = \left(x - 3\right)^{4}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = x - 3$.

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

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

      1. Differentiate $x - 3$ term by term:

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

         2. The derivative of the constant $-3$ is zero.

         The result is: $1$

      The result of the chain rule is:

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

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

2. Now simplify:

   $$\left(x - 3\right)^{3} \left(x + 2\right)^{4} \left(9 x - 7\right)$$

The answer is:

$$\left(x - 3\right)^{3} \left(x + 2\right)^{4} \left(9 x - 7\right)$$