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

1. Apply the product rule:

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

   $f{\left(x \right)} = \left(3 x + 4\right)^{2}$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Let $u = 3 x + 4$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(3 x + 4\right)$:

      1. Differentiate $3 x + 4$ 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: $3$

         2. The derivative of the constant $4$ is zero.

         The result is: $3$

      The result of the chain rule is:

      $$18 x + 24$$

   $g{\left(x \right)} = \left(x - 5\right)^{3}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = x - 5$.

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

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

      1. Differentiate $x - 5$ term by term:

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

         2. The derivative of the constant $\left(-1\right) 5$ is zero.

         The result is: $1$

      The result of the chain rule is:

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

   The result is: $\left(x - 5\right)^{3} \cdot \left(18 x + 24\right) + 3 \left(x - 5\right)^{2} \left(3 x + 4\right)^{2}$

2. Now simplify:

   $$3 \left(x - 5\right)^{2} \cdot \left(3 x + 4\right) \left(5 x - 6\right)$$

The answer is:

$$3 \left(x - 5\right)^{2} \cdot \left(3 x + 4\right) \left(5 x - 6\right)$$