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Derivative of:
Derivative of (sin(x))^2
Derivative of x-1
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Derivative of sin(x)/x^2
Identical expressions
(x- one)/(ln(x- one)/(x- one))
(x minus 1) divide by (ln(x minus 1) divide by (x minus 1))
(x minus one) divide by (ln(x minus one) divide by (x minus one))
x-1/lnx-1/x-1
(x-1) divide by (ln(x-1) divide by (x-1))
Similar expressions
(x-1)/(ln(x-1)/(x+1))
(x-1)/(ln(x+1)/(x-1))
(x+1)/(ln(x-1)/(x-1))

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

Derivative of (x-1)/(ln(x-1)/(x-1))

The solution

You have entered [src]

   x - 1    
------------
/log(x - 1)\
|----------|
\  x - 1   /

\frac{x - 1}{\frac{1}{x - 1} \log{\left(x - 1 \right)}}

(x - 1)/((log(x - 1)/(x - 1)))

Detail solution

Apply the quotient rule, which is:

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

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                      3 /     1       log(x - 1)\
               (x - 1) *|- -------- + ----------|
                        |         2           2 |
     1                  \  (x - 1)     (x - 1)  /
------------ + ----------------------------------
/log(x - 1)\                 2                   
|----------|              log (x - 1)            
\  x - 1   /

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

Simplify

The second derivative [src]

/         1     \                      -1 + log(-1 + x)              
|1 - -----------|*(-1 + log(-1 + x)) - ---------------- + log(-1 + x)
\    log(-1 + x)/                        log(-1 + x)                 
---------------------------------------------------------------------
                                2                                    
                             log (-1 + x)

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

Simplify

The third derivative [src]

                                                                                                                                                                             /         1     \                      /         2     \                   
                                                                                                                                                                             |1 - -----------|*(-1 + log(-1 + x))   |1 - -----------|*(-1 + log(-1 + x))
                    /         1     \                        6*(-1 + log(-1 + x))   3*(-3 + 2*log(-1 + x))   3*(-1 + log(-1 + x))     /         1     \                      \    log(-1 + x)/                      \    log(-1 + x)/                   
3 - 2*log(-1 + x) - |1 - -----------|*(-3 + 2*log(-1 + x)) - -------------------- + ---------------------- + -------------------- + 4*|1 - -----------|*(-1 + log(-1 + x)) - ------------------------------------ - ------------------------------------
                    \    log(-1 + x)/                            log(-1 + x)             log(-1 + x)                2                 \    log(-1 + x)/                                  log(-1 + x)                            log(-1 + x)             
                                                                                                                 log (-1 + x)                                                                                                                           
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                             2                                                                                                                          
                                                                                                                 (-1 + x)*log (-1 + x)

\frac{- \frac{\left(1 - \frac{2}{\log{\left(x - 1 \right)}}\right) \left(\log{\left(x - 1 \right)} - 1\right)}{\log{\left(x - 1 \right)}} + 4 \left(1 - \frac{1}{\log{\left(x - 1 \right)}}\right) \left(\log{\left(x - 1 \right)} - 1\right) - \frac{\left(1 - \frac{1}{\log{\left(x - 1 \right)}}\right) \left(\log{\left(x - 1 \right)} - 1\right)}{\log{\left(x - 1 \right)}} - \left(1 - \frac{1}{\log{\left(x - 1 \right)}}\right) \left(2 \log{\left(x - 1 \right)} - 3\right) - \frac{6 \left(\log{\left(x - 1 \right)} - 1\right)}{\log{\left(x - 1 \right)}} + \frac{3 \left(\log{\left(x - 1 \right)} - 1\right)}{\log{\left(x - 1 \right)}^{2}} + \frac{3 \left(2 \log{\left(x - 1 \right)} - 3\right)}{\log{\left(x - 1 \right)}} - 2 \log{\left(x - 1 \right)} + 3}{\left(x - 1\right) \log{\left(x - 1 \right)}^{2}}

Simplify

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Derivative of (x-1)/(ln(x-1)/(x-1))

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The solution