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The derivative of the function/ (3x^4-4x^3)/(5x+1)

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

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  /

$$\frac{d}{d x} \frac{3 x^{4} - 4 x^{3}}{5 x + 1}$$

Transform

Detail solution

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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      2       3     /   4      3\
- 12*x  + 12*x    5*\3*x  - 4*x /
--------------- - ---------------
    5*x + 1                   2  
                     (5*x + 1)

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

Simplify

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

$$\frac{2 x \left(\frac{25 x^{2} \cdot \left(3 x - 4\right)}{\left(5 x + 1\right)^{2}} - \frac{60 x \left(x - 1\right)}{5 x + 1} + 18 x - 12\right)}{5 x + 1}$$

Simplify

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

$$\frac{6 \left(- \frac{125 x^{3} \cdot \left(3 x - 4\right)}{\left(5 x + 1\right)^{3}} + \frac{300 x^{2} \left(x - 1\right)}{\left(5 x + 1\right)^{2}} - \frac{30 x \left(3 x - 2\right)}{5 x + 1} + 12 x - 4\right)}{5 x + 1}$$

Simplify

The graph

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

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