Derivative of y=xsqrt(1-x^3)

1. Apply the product rule:

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

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

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

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

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

   2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$

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

      1. Differentiate $1 - x^{3}$ term by term:

         1. The derivative of the constant $1$ is zero.

         2. 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: $- 3 x^{2}$

         The result is: $- 3 x^{2}$

      The result of the chain rule is:

      $$- \frac{3 x^{2}}{2 \sqrt{1 - x^{3}}}$$

   The result is: $- \frac{3 x^{3}}{2 \sqrt{1 - x^{3}}} + \sqrt{1 - x^{3}}$

2. Now simplify:

   $$\frac{2 - 5 x^{3}}{2 \sqrt{1 - x^{3}}}$$

The answer is:

$$\frac{2 - 5 x^{3}}{2 \sqrt{1 - x^{3}}}$$