Derivative of (3x^4-4x^3)/(5x+1)

1. Apply the quotient rule, which is:

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

   $f{\left(x \right)} = 3 x^{4} - 4 x^{3}$ and $g{\left(x \right)} = 5 x + 1$.

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

   1. Differentiate $3 x^{4} - 4 x^{3}$ 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: $- 12 x^{2}$

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

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

         So, the result is: $12 x^{3}$

      The result is: $12 x^{3} - 12 x^{2}$

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

   1. Differentiate $5 x + 1$ 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: $x$ goes to $1$

         So, the result is: $5$

      The result is: $5$

   Now plug in to the quotient rule:

   $\frac{- 15 x^{4} + 20 x^{3} + \left(5 x + 1\right) \left(12 x^{3} - 12 x^{2}\right)}{\left(5 x + 1\right)^{2}}$

2. Now simplify:

   $$\frac{x^{2} \cdot \left(45 x^{2} - 28 x - 12\right)}{25 x^{2} + 10 x + 1}$$

The answer is:

$$\frac{x^{2} \cdot \left(45 x^{2} - 28 x - 12\right)}{25 x^{2} + 10 x + 1}$$