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Identical expressions
y=(x- one)(x²+x+ one)/(x³)
y equally (x minus 1)(x² plus x plus 1) divide by (x³)
y equally (x minus one)(x² plus x plus one) divide by (x³)
y=x-1x²+x+1/x³
y=(x-1)(x²+x+1) divide by (x³)
Similar expressions
y=(x-1)(x²-x+1)/(x³)
y=(x-1)(x²+x-1)/(x³)
y=(x+1)(x²+x+1)/(x³)

The derivative of the function/ y=(x-1)(x²+x+1)/(x³)

Derivative of y=(x-1)(x²+x+1)/(x³)

The solution

You have entered [src]

        / 2        \
(x - 1)*\x  + x + 1/
--------------------
          3         
         x

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

((x - 1)*(x^2 + x + 1))/x^3

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \left(x - 1\right) \left(x^{2} + x + 1\right)$ and $g{\left(x \right)} = x^{3}$ .

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

$\frac{x^{3} \left(x^{2} + x + \left(x - 1\right) \left(2 x + 1\right) + 1\right) - 3 x^{2} \left(x - 1\right) \left(x^{2} + x + 1\right)}{x^{6}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of y=(x-1)(x²+x+1)/(x³)

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The solution