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

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

Derivative of 1/(x^3+1)

The solution

You have entered [src]

    1   
1*------
   3    
  x  + 1

$$1 \cdot \frac{1}{x^{3} + 1}$$

d /    1   \
--|1*------|
dx|   3    |
  \  x  + 1/

$$\frac{d}{d x} 1 \cdot \frac{1}{x^{3} + 1}$$

Transform

Detail solution

Apply the quotient rule, which is:

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

   /        3          6  \
   |    18*x       27*x   |
-6*|1 - ------ + ---------|
   |         3           2|
   |    1 + x    /     3\ |
   \             \1 + x / /
---------------------------
                 2         
         /     3\          
         \1 + x /

$$- \frac{6 \cdot \left(\frac{27 x^{6}}{\left(x^{3} + 1\right)^{2}} - \frac{18 x^{3}}{x^{3} + 1} + 1\right)}{\left(x^{3} + 1\right)^{2}}$$

Simplify

The graph

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Derivative of 1/(x^3+1)

The graph:

Piecewise:

The solution