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

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

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
   4      3
3*x  - 4*x 
-----------
  5*x + 1  
$$\frac{3 x^{4} - 4 x^{3}}{5 x + 1}$$
  /   4      3\
d |3*x  - 4*x |
--|-----------|
dx\  5*x + 1  /
$$\frac{d}{d x} \frac{3 x^{4} - 4 x^{3}}{5 x + 1}$$
Detail solution
  1. Apply the quotient rule, which is:

    and .

    To find :

    1. Differentiate 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: goes to

        So, the result is:

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

        1. Apply the power rule: goes to

        So, the result is:

      The result is:

    To find :

    1. Differentiate term by term:

      1. The derivative of the constant 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: goes to

        So, the result is:

      The result is:

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
      2       3     /   4      3\
- 12*x  + 12*x    5*\3*x  - 4*x /
--------------- - ---------------
    5*x + 1                   2  
                     (5*x + 1)   
$$\frac{12 x^{3} - 12 x^{2}}{5 x + 1} - \frac{5 \cdot \left(3 x^{4} - 4 x^{3}\right)}{\left(5 x + 1\right)^{2}}$$
The second derivative [src]
    /                                 2           \
    |             60*x*(-1 + x)   25*x *(-4 + 3*x)|
2*x*|-12 + 18*x - ------------- + ----------------|
    |                1 + 5*x                  2   |
    \                                (1 + 5*x)    /
---------------------------------------------------
                      1 + 5*x                      
$$\frac{2 x \left(\frac{25 x^{2} \cdot \left(3 x - 4\right)}{\left(5 x + 1\right)^{2}} - \frac{60 x \left(x - 1\right)}{5 x + 1} + 18 x - 12\right)}{5 x + 1}$$
The third derivative [src]
  /                 3                                     2         \
  |            125*x *(-4 + 3*x)   30*x*(-2 + 3*x)   300*x *(-1 + x)|
6*|-4 + 12*x - ----------------- - --------------- + ---------------|
  |                         3          1 + 5*x                   2  |
  \                (1 + 5*x)                            (1 + 5*x)   /
---------------------------------------------------------------------
                               1 + 5*x                               
$$\frac{6 \left(- \frac{125 x^{3} \cdot \left(3 x - 4\right)}{\left(5 x + 1\right)^{3}} + \frac{300 x^{2} \left(x - 1\right)}{\left(5 x + 1\right)^{2}} - \frac{30 x \left(3 x - 2\right)}{5 x + 1} + 12 x - 4\right)}{5 x + 1}$$
The graph
Derivative of (3x^4-4x^3)/(5x+1)