Derivative of 1/x^3-1/x^4

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

   1. Apply the quotient rule, which is:

      $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

      $f{\left(x \right)} = 1$ and $g{\left(x \right)} = x^{3}$.

      To find $\frac{d}{d x} f{\left(x \right)}$:

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

      To find $\frac{d}{d x} g{\left(x \right)}$:

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

      Now plug in to the quotient rule:

      $- \frac{3}{x^{4}}$

   2. 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: $\frac{4}{x^{5}}$

   The result is: $- \frac{3}{x^{4}} + \frac{4}{x^{5}}$

2. Now simplify:

   $$\frac{4 - 3 x}{x^{5}}$$

The answer is:

$$\frac{4 - 3 x}{x^{5}}$$