Derivative of y=x*x^2*(x^2)^3

1. Apply the product rule:

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

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

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

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

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

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

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

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} x^{2}$:

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

      The result of the chain rule is:

      $$6 x^{5}$$

   The result is: $6 x^{8} + 3 x^{2} x^{6}$

2. Now simplify:

   $$9 x^{8}$$

The answer is:

$$9 x^{8}$$