Derivative of (3x²+4x-5)³

1. Let $u = \left(3 x^{2} + 4 x\right) - 5$.

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

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\left(3 x^{2} + 4 x\right) - 5\right)$:

   1. Differentiate $\left(3 x^{2} + 4 x\right) - 5$ term by term:

      1. Differentiate $3 x^{2} + 4 x$ 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: $x^{2}$ goes to $2 x$

            So, the result is: $6 x$

         2. 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: $4$

         The result is: $6 x + 4$

      2. The derivative of the constant $-5$ is zero.

      The result is: $6 x + 4$

   The result of the chain rule is:

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

4. Now simplify:

   $$\left(18 x + 12\right) \left(3 x^{2} + 4 x - 5\right)^{2}$$

The answer is:

$$\left(18 x + 12\right) \left(3 x^{2} + 4 x - 5\right)^{2}$$