Derivative of y=x⁹-3x⁵-3:x⁴+2

1. Differentiate $x^{9} - 3 x^{5} + 2 - \frac{3}{x^{4}}$ term by term:

   1. Apply the power rule: $x^{9}$ goes to $9 x^{8}$

   2. The derivative of a constant times a function is the constant times the derivative of the function.

      1. The derivative of a constant times a function is the constant times the derivative of the function.

         1. Apply the power rule: $x^{5}$ goes to $5 x^{4}$

         So, the result is: $15 x^{4}$

      So, the result is: $- 15 x^{4}$

   3. The derivative of a constant times a function is the constant times the derivative of the function.

      1. The derivative of a constant times a function is the constant times the derivative of the function.

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

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

         3. Then, apply the chain rule. Multiply by $\frac{d}{d x} x^{4}$:

            1. Apply the power rule: $x^{4}$ goes to $4 x^{3}$

            The result of the chain rule is:

            $$- \frac{4}{x^{5}}$$

         So, the result is: $- \frac{12}{x^{5}}$

      So, the result is: $\frac{12}{x^{5}}$

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

   The result is: $9 x^{8} - 15 x^{4} + \frac{12}{x^{5}}$

2. Now simplify:

   $$\frac{3 \cdot \left(3 x^{13} - 5 x^{9} + 4\right)}{x^{5}}$$

The answer is:

$$\frac{3 \cdot \left(3 x^{13} - 5 x^{9} + 4\right)}{x^{5}}$$