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The derivative of the function/ x/√(x-1)

Derivative of x/√(x-1)

The solution

You have entered [src]

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

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

x/sqrt(x - 1)

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
To find $\frac{d}{d x} g{\left(x \right)}$ :
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:
  
  $\frac{1}{2 \sqrt{x - 1}}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    1            x      
--------- - ------------
  _______            3/2
\/ x - 1    2*(x - 1)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /     5*x  \
3*|6 - ------|
  \    -1 + x/
--------------
          5/2 
8*(-1 + x)

$$\frac{3 \left(- \frac{5 x}{x - 1} + 6\right)}{8 \left(x - 1\right)^{\frac{5}{2}}}$$

Simplify

The graph

Derivative of x/√(x-1)