Mister Exam

Other calculators

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

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

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
       1       
---------------
   ____________
  /          3 
\/  1 + x - x
1−x3+(x+1)\frac{1}{\sqrt{- x^{3} + \left(x + 1\right)}} 
1/(sqrt(1 + x - x^3))
Detail solution

  1. Let u=−x3+(x+1)u = \sqrt{- x^{3} + \left(x + 1\right)}.

  2. Apply the power rule: 1u\frac{1}{u} goes to −1u2- \frac{1}{u^{2}}

  3. Then, apply the chain rule. Multiply by ddx−x3+(x+1)\frac{d}{d x} \sqrt{- x^{3} + \left(x + 1\right)}:

    1. Let u=−x3+(x+1)u = - x^{3} + \left(x + 1\right).

    2. Apply the power rule: u\sqrt{u} goes to 12u\frac{1}{2 \sqrt{u}}

    3. Then, apply the chain rule. Multiply by ddx(−x3+(x+1))\frac{d}{d x} \left(- x^{3} + \left(x + 1\right)\right):

      1. Differentiate −x3+(x+1)- x^{3} + \left(x + 1\right) term by term:

        1. Differentiate x+1x + 1 term by term:

          1. The derivative of the constant 11 is zero.

          2. Apply the power rule: xx goes to 11

          The result is: 11

        2. The derivative of a constant times a function is the constant times the derivative of the function.

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

          So, the result is: −3x2- 3 x^{2}

        The result is: 1−3x21 - 3 x^{2}

      The result of the chain rule is:

      1−3x22−x3+(x+1)\frac{1 - 3 x^{2}}{2 \sqrt{- x^{3} + \left(x + 1\right)}}

    The result of the chain rule is:

    −1−3x22(−x3+(x+1))32- \frac{1 - 3 x^{2}}{2 \left(- x^{3} + \left(x + 1\right)\right)^{\frac{3}{2}}}

  4. Now simplify:

    3x2−12(−x3+x+1)32\frac{3 x^{2} - 1}{2 \left(- x^{3} + x + 1\right)^{\frac{3}{2}}}

The answer is:

3x2−12(−x3+x+1)32\frac{3 x^{2} - 1}{2 \left(- x^{3} + x + 1\right)^{\frac{3}{2}}}

The graph
02468-8-6-4-2-1010-200200
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
−12−3x22−x3+(x+1)(−x3+(x+1))- \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 /
3(x+(3x2−1)24(−x3+x+1))(−x3+x+1)32\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 /
3(9x(3x2−1)2(−x3+x+1)+5(3x2−1)38(−x3+x+1)2+1)(−x3+x+1)32\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
    © Mister Exam – Calculator