Derivative of root(x-1)/x

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  _______
\/ x - 1 
---------
    x

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

sqrt(x - 1)/x

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 $g{\left(x \right)} = x$ .

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. Apply the power rule: $x$ goes to $1$
Now plug in to the quotient rule:

$\frac{\frac{x}{2 \sqrt{x - 1}} - \sqrt{x - 1}}{x^{2}}$
Now simplify:

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

The answer is:

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

The graph

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The first derivative [src]

                  _______
      1         \/ x - 1 
------------- - ---------
      _______        2   
2*x*\/ x - 1        x

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of root(x-1)/x

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The solution