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

Derivative of 1/(x^3+1)

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
    1   
1*------
   3    
  x  + 1
1⋅1x3+11 \cdot \frac{1}{x^{3} + 1} 
d /    1   \
--|1*------|
dx|   3    |
  \  x  + 1/
ddx1⋅1x3+1\frac{d}{d x} 1 \cdot \frac{1}{x^{3} + 1}
Transform
Detail solution

  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=1f{\left(x \right)} = 1 and g(x)=x3+1g{\left(x \right)} = x^{3} + 1.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. The derivative of the constant 11 is zero.

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Differentiate x3+1x^{3} + 1 term by term:

      1. The derivative of the constant 11 is zero.

      2. Apply the power rule: x3x^{3} goes to 3x23 x^{2}

      The result is: 3x23 x^{2}

    Now plug in to the quotient rule:

    −3x2(x3+1)2- \frac{3 x^{2}}{\left(x^{3} + 1\right)^{2}}

The answer is:

−3x2(x3+1)2- \frac{3 x^{2}}{\left(x^{3} + 1\right)^{2}}

The graph
02468-8-6-4-2-1010-5050
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
      2  
  -3*x   
---------
        2
/ 3    \ 
\x  + 1/
−3x2(x3+1)2- \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 /
6x(3x3x3+1−1)(x3+1)2\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 /
−6⋅(27x6(x3+1)2−18x3x3+1+1)(x3+1)2- \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
Derivative of 1/(x^3+1)
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