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Identical expressions
(x^ five -x^ three + one)/(x- one)
(x to the power of 5 minus x cubed plus 1) divide by (x minus 1)
(x to the power of five minus x to the power of three plus one) divide by (x minus one)
(x5-x3+1)/(x-1)
x5-x3+1/x-1
(x⁵-x³+1)/(x-1)
(x to the power of 5-x to the power of 3+1)/(x-1)
x^5-x^3+1/x-1
(x^5-x^3+1) divide by (x-1)
Similar expressions
(x^5+x^3+1)/(x-1)
(x^5-x^3-1)/(x-1)
(x^5-x^3+1)/(x+1)

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

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

The solution

You have entered [src]

 5    3    
x  - x  + 1
-----------
   x - 1

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

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

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x^{5} - x^{3} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Apply the power rule: $x^{5}$ goes to $5 x^{4}$
  3. 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: $- 3 x^{2}$
  The result is: $5 x^{4} - 3 x^{2}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x - 1$ term by term:
  1. The derivative of the constant $-1$ is zero.
  2. Apply the power rule: $x$ goes to $1$
  The result is: $1$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     2      4    5    3    
- 3*x  + 5*x    x  - x  + 1
------------- - -----------
    x - 1                2 
                  (x - 1)

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

Simplify

The second derivative [src]

  /                      5    3    2 /        2\\
  |  /         2\   1 + x  - x    x *\-3 + 5*x /|
2*|x*\-3 + 10*x / + ----------- - --------------|
  |                          2        -1 + x    |
  \                  (-1 + x)                   /
-------------------------------------------------
                      -1 + x

$$\frac{2 \left(- \frac{x^{2} \left(5 x^{2} - 3\right)}{x - 1} + x \left(10 x^{2} - 3\right) + \frac{x^{5} - x^{3} + 1}{\left(x - 1\right)^{2}}\right)}{x - 1}$$

Simplify

The third derivative [src]

  /                  5    3    2 /        2\     /         2\\
  |         2   1 + x  - x    x *\-3 + 5*x /   x*\-3 + 10*x /|
6*|-1 + 10*x  - ----------- + -------------- - --------------|
  |                      3              2          -1 + x    |
  \              (-1 + x)       (-1 + x)                     /
--------------------------------------------------------------
                            -1 + x

$$\frac{6 \left(10 x^{2} + \frac{x^{2} \left(5 x^{2} - 3\right)}{\left(x - 1\right)^{2}} - \frac{x \left(10 x^{2} - 3\right)}{x - 1} - 1 - \frac{x^{5} - x^{3} + 1}{\left(x - 1\right)^{3}}\right)}{x - 1}$$

Simplify

The graph

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

The graph:

Piecewise:

The solution