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one /sqrt(one +x-x^ three)
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Similar expressions
1/sqrt(1-x-x^3)
1/sqrt(1+x+x^3)

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

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

The solution

You have entered [src]

       1       
---------------
   ____________
  /          3 
\/  1 + x - x

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

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

Detail solution

Let $u = \sqrt{- x^{3} + \left(x + 1\right)}$ .
Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{- x^{3} + \left(x + 1\right)}$ :
1. Let $u = - x^{3} + \left(x + 1\right)$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(- x^{3} + \left(x + 1\right)\right)$ :
  1. Differentiate $- x^{3} + \left(x + 1\right)$ term by term:
    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: $1$
    2. 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: $1 - 3 x^{2}$
  The result of the chain rule is:
  
  $\frac{1 - 3 x^{2}}{2 \sqrt{- x^{3} + \left(x + 1\right)}}$
The result of the chain rule is:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  /                2 \
  |     /        2\  |
  |     \-1 + 3*x /  |
3*|x + --------------|
  |      /         3\|
  \    4*\1 + x - x //
----------------------
               3/2    
   /         3\       
   \1 + x - x /

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

Simplify

The third derivative [src]

  /                  3                  \
  |       /        2\        /        2\|
  |     5*\-1 + 3*x /    9*x*\-1 + 3*x /|
3*|1 + --------------- + ---------------|
  |                  2      /         3\|
  |      /         3\     2*\1 + x - x /|
  \    8*\1 + x - x /                   /
-----------------------------------------
                         3/2             
             /         3\                
             \1 + x - x /

$$\frac{3 \left(\frac{9 x \left(3 x^{2} - 1\right)}{2 \left(- x^{3} + x + 1\right)} + \frac{5 \left(3 x^{2} - 1\right)^{3}}{8 \left(- x^{3} + x + 1\right)^{2}} + 1\right)}{\left(- x^{3} + x + 1\right)^{\frac{3}{2}}}$$

Simplify

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

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The solution