Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of (x+1)/x
Derivative of sqrt(x-1)/(sqrt(x^2-x)-1)
Derivative of 4*sin(x)-6*x+7
Derivative of 3x-ln(x+3)^3
Identical expressions
sqrt(x- one)/(sqrt(x^ two -x)- one)
square root of (x minus 1) divide by ( square root of (x squared minus x) minus 1)
square root of (x minus one) divide by ( square root of (x to the power of two minus x) minus one)
√(x-1)/(√(x^2-x)-1)
sqrt(x-1)/(sqrt(x2-x)-1)
sqrtx-1/sqrtx2-x-1
sqrt(x-1)/(sqrt(x²-x)-1)
sqrt(x-1)/(sqrt(x to the power of 2-x)-1)
sqrtx-1/sqrtx^2-x-1
sqrt(x-1) divide by (sqrt(x^2-x)-1)
Similar expressions
(sqrt(x)-1)/sqrt(x^2-x-1)
sqrt(x-1)/(sqrt(x^2-x)+1)
sqrt(x-1)/(sqrt(x^2+x)-1)
sqrt(x+1)/(sqrt(x^2-x)-1)

The derivative of the function/ sqrt(x-1)/(sqrt(x^2-x)-1)

Derivative of sqrt(x-1)/(sqrt(x^2-x)-1)

The solution

You have entered [src]

     _______   
   \/ x - 1    
---------------
   ________    
  /  2         
\/  x  - x  - 1

$$\frac{\sqrt{x - 1}}{\sqrt{x^{2} - x} - 1}$$

sqrt(x - 1)/(sqrt(x^2 - x) - 1)

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = x - 1$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - 1\right)$ :
  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: $1$
  The result of the chain rule is:
  
  $\frac{1}{2 \sqrt{x - 1}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $\sqrt{x^{2} - x} - 1$ term by term:
  1. The derivative of the constant $-1$ is zero.
  2. Let $u = x^{2} - x$ .
  3. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
  4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} - x\right)$ :
    1. Differentiate $x^{2} - x$ term by term:
      1. Apply the power rule: $x^{2}$ goes to $2 x$
      2. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $-1$
      The result is: $2 x - 1$
    The result of the chain rule is:
    
    $\frac{2 x - 1}{2 \sqrt{x^{2} - x}}$
  The result is: $\frac{2 x - 1}{2 \sqrt{x^{2} - x}}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                                       _______                
              1                      \/ x - 1 *(-1/2 + x)     
----------------------------- - ------------------------------
            /   ________    \                                2
    _______ |  /  2         |      ________ /   ________    \ 
2*\/ x - 1 *\\/  x  - x  - 1/     /  2      |  /  2         | 
                                \/  x  - x *\\/  x  - x  - 1/

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                                                                                                                                                               /                                2                                                                2                                      2           \\
  |                                             2                          2                                                                  ________            |         4            (-1 + 2*x)                     8                               2*(-1 + 2*x)                           2*(-1 + 2*x)            ||
  |                      4            (-1 + 2*x)               2*(-1 + 2*x)                                                                 \/ -1 + x *(-1 + 2*x)*|- --------------- + --------------- - -------------------------------- + -------------------------------------- + ----------------------------------||
  |              - -------------- + --------------- + --------------------------------                                                                            |              3/2               5/2              /       ____________\                                        2    2         2 /       ____________\||
  |                  ____________               3/2              /       ____________\                                                                            |  (x*(-1 + x))      (x*(-1 + x))      x*(-1 + x)*\-1 + \/ x*(-1 + x) /               3/2 /       ____________\    x *(-1 + x) *\-1 + \/ x*(-1 + x) /||
  |     1          \/ x*(-1 + x)    (x*(-1 + x))      x*(-1 + x)*\-1 + \/ x*(-1 + x) /                       -1 + 2*x                                             \                                                                         (x*(-1 + x))   *\-1 + \/ x*(-1 + x) /                                      /|
3*|----------- + --------------------------------------------------------------------- + ------------------------------------------------ - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
  |        5/2                        ________ /       ____________\                       ____________         3/2 /       ____________\                                                                                      ____________                                                                             |
  \(-1 + x)                         \/ -1 + x *\-1 + \/ x*(-1 + x) /                     \/ x*(-1 + x) *(-1 + x)   *\-1 + \/ x*(-1 + x) /                                                                               -1 + \/ x*(-1 + x)                                                                              /
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                   /       ____________\                                                                                                                                                 
                                                                                                                                                 8*\-1 + \/ x*(-1 + x) /

$$\frac{3 \left(- \frac{\sqrt{x - 1} \left(2 x - 1\right) \left(\frac{2 \left(2 x - 1\right)^{2}}{\left(x \left(x - 1\right)\right)^{\frac{3}{2}} \left(\sqrt{x \left(x - 1\right)} - 1\right)^{2}} - \frac{4}{\left(x \left(x - 1\right)\right)^{\frac{3}{2}}} + \frac{\left(2 x - 1\right)^{2}}{\left(x \left(x - 1\right)\right)^{\frac{5}{2}}} - \frac{8}{x \left(x - 1\right) \left(\sqrt{x \left(x - 1\right)} - 1\right)} + \frac{2 \left(2 x - 1\right)^{2}}{x^{2} \left(x - 1\right)^{2} \left(\sqrt{x \left(x - 1\right)} - 1\right)}\right)}{\sqrt{x \left(x - 1\right)} - 1} + \frac{- \frac{4}{\sqrt{x \left(x - 1\right)}} + \frac{\left(2 x - 1\right)^{2}}{\left(x \left(x - 1\right)\right)^{\frac{3}{2}}} + \frac{2 \left(2 x - 1\right)^{2}}{x \left(x - 1\right) \left(\sqrt{x \left(x - 1\right)} - 1\right)}}{\sqrt{x - 1} \left(\sqrt{x \left(x - 1\right)} - 1\right)} + \frac{1}{\left(x - 1\right)^{\frac{5}{2}}} + \frac{2 x - 1}{\sqrt{x \left(x - 1\right)} \left(x - 1\right)^{\frac{3}{2}} \left(\sqrt{x \left(x - 1\right)} - 1\right)}\right)}{8 \left(\sqrt{x \left(x - 1\right)} - 1\right)}$$

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of sqrt(x-1)/(sqrt(x^2-x)-1)

The graph:

Piecewise:

The solution