Derivative of y=(x²+4)²(2x³-1)³

1. Apply the product rule:

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

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

   1. Let $u = x^{2} + 4$.

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

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

      1. Differentiate $x^{2} + 4$ term by term:

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

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

         The result is: $2 x$

      The result of the chain rule is:

      $$2 x \left(2 x^{2} + 8\right)$$

   $g{\left(x \right)} = \left(2 x^{3} - 1\right)^{3}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = 2 x^{3} - 1$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x^{3} - 1\right)$:

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

            So, the result is: $6 x^{2}$

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

         The result is: $6 x^{2}$

      The result of the chain rule is:

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

   The result is: $18 x^{2} \left(x^{2} + 4\right)^{2} \left(2 x^{3} - 1\right)^{2} + 2 x \left(2 x^{2} + 8\right) \left(2 x^{3} - 1\right)^{3}$

2. Now simplify:

   $$2 x \left(x^{2} + 4\right) \left(2 x^{3} - 1\right)^{2} \cdot \left(4 x^{3} + 9 x \left(x^{2} + 4\right) - 2\right)$$

The answer is: