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Derivative of:
Derivative of (sin(x))^2
Derivative of x-1
Derivative of e^(7*x)
Derivative of sin(x)/x^2
Identical expressions
ctg(ln(four ^x- one))
ctg(ln(4 to the power of x minus 1))
ctg(ln(four to the power of x minus one))
ctg(ln(4x-1))
ctgln4x-1
ctgln4^x-1
Similar expressions
ctg(ln(4^x+1))

The derivative of the function/ ctg(ln(4^x-1))

Derivative of ctg(ln(4^x-1))

The solution

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   /   / x    \\
cot\log\4  - 1//

$$\cot{\left(\log{\left(4^{x} - 1 \right)} \right)}$$

cot(log(4^x - 1))

Detail solution

There are multiple ways to do this derivative.
Method #1
1. Rewrite the function to be differentiated:
  
  $\cot{\left(\log{\left(4^{x} - 1 \right)} \right)} = \frac{1}{\tan{\left(\log{\left(4^{x} - 1 \right)} \right)}}$
2. Let $u = \tan{\left(\log{\left(4^{x} - 1 \right)} \right)}$ .
3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(\log{\left(4^{x} - 1 \right)} \right)}$ :
  1. Rewrite the function to be differentiated:
    
    $\tan{\left(\log{\left(4^{x} - 1 \right)} \right)} = \frac{\sin{\left(\log{\left(4^{x} - 1 \right)} \right)}}{\cos{\left(\log{\left(4^{x} - 1 \right)} \right)}}$
  2. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \sin{\left(\log{\left(4^{x} - 1 \right)} \right)}$ and $g{\left(x \right)} = \cos{\left(\log{\left(4^{x} - 1 \right)} \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Let $u = \log{\left(4^{x} - 1 \right)}$ .
    2. The derivative of sine is cosine:
      
      $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(4^{x} - 1 \right)}$ :
      
      Let $u = 4^{x} - 1$ .
      
      The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(4^{x} - 1\right)$ :
      
      Differentiate $4^{x} - 1$ term by term:
      
      $\frac{d}{d x} 4^{x} = 4^{x} \log{\left(4 \right)}$
      
      The derivative of the constant $-1$ is zero.
      
      The result is: $4^{x} \log{\left(4 \right)}$
      
      The result of the chain rule is:
      
      $\frac{4^{x} \log{\left(4 \right)}}{4^{x} - 1}$
      
      The result of the chain rule is:
      
      $\frac{4^{x} \log{\left(4 \right)} \cos{\left(\log{\left(4^{x} - 1 \right)} \right)}}{4^{x} - 1}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Let $u = \log{\left(4^{x} - 1 \right)}$ .
    2. The derivative of cosine is negative sine:
      
      $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(4^{x} - 1 \right)}$ :
      
      Let $u = 4^{x} - 1$ .
      
      The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(4^{x} - 1\right)$ :
      
      Differentiate $4^{x} - 1$ term by term:
      
      $\frac{d}{d x} 4^{x} = 4^{x} \log{\left(4 \right)}$
      
      The derivative of the constant $-1$ is zero.
      
      The result is: $4^{x} \log{\left(4 \right)}$
      
      The result of the chain rule is:
      
      $\frac{4^{x} \log{\left(4 \right)}}{4^{x} - 1}$
      
      The result of the chain rule is:
      
      $- \frac{4^{x} \log{\left(4 \right)} \sin{\left(\log{\left(4^{x} - 1 \right)} \right)}}{4^{x} - 1}$
    Now plug in to the quotient rule:
    
    $\frac{\frac{4^{x} \log{\left(4 \right)} \sin^{2}{\left(\log{\left(4^{x} - 1 \right)} \right)}}{4^{x} - 1} + \frac{4^{x} \log{\left(4 \right)} \cos^{2}{\left(\log{\left(4^{x} - 1 \right)} \right)}}{4^{x} - 1}}{\cos^{2}{\left(\log{\left(4^{x} - 1 \right)} \right)}}$
  The result of the chain rule is:
  
  $- \frac{\frac{4^{x} \log{\left(4 \right)} \sin^{2}{\left(\log{\left(4^{x} - 1 \right)} \right)}}{4^{x} - 1} + \frac{4^{x} \log{\left(4 \right)} \cos^{2}{\left(\log{\left(4^{x} - 1 \right)} \right)}}{4^{x} - 1}}{\cos^{2}{\left(\log{\left(4^{x} - 1 \right)} \right)} \tan^{2}{\left(\log{\left(4^{x} - 1 \right)} \right)}}$
Method #2
1. Rewrite the function to be differentiated:
  
  $\cot{\left(\log{\left(4^{x} - 1 \right)} \right)} = \frac{\cos{\left(\log{\left(4^{x} - 1 \right)} \right)}}{\sin{\left(\log{\left(4^{x} - 1 \right)} \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \cos{\left(\log{\left(4^{x} - 1 \right)} \right)}$ and $g{\left(x \right)} = \sin{\left(\log{\left(4^{x} - 1 \right)} \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = \log{\left(4^{x} - 1 \right)}$ .
  2. The derivative of cosine is negative sine:
    
    $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(4^{x} - 1 \right)}$ :
    1. Let $u = 4^{x} - 1$ .
    2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(4^{x} - 1\right)$ :
      
      Differentiate $4^{x} - 1$ term by term:
      
      $\frac{d}{d x} 4^{x} = 4^{x} \log{\left(4 \right)}$
      
      The derivative of the constant $-1$ is zero.
      
      The result is: $4^{x} \log{\left(4 \right)}$
      
      The result of the chain rule is:
      
      $\frac{4^{x} \log{\left(4 \right)}}{4^{x} - 1}$
    The result of the chain rule is:
    
    $- \frac{4^{x} \log{\left(4 \right)} \sin{\left(\log{\left(4^{x} - 1 \right)} \right)}}{4^{x} - 1}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = \log{\left(4^{x} - 1 \right)}$ .
  2. The derivative of sine is cosine:
    
    $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(4^{x} - 1 \right)}$ :
    1. Let $u = 4^{x} - 1$ .
    2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(4^{x} - 1\right)$ :
      
      Differentiate $4^{x} - 1$ term by term:
      
      $\frac{d}{d x} 4^{x} = 4^{x} \log{\left(4 \right)}$
      
      The derivative of the constant $-1$ is zero.
      
      The result is: $4^{x} \log{\left(4 \right)}$
      
      The result of the chain rule is:
      
      $\frac{4^{x} \log{\left(4 \right)}}{4^{x} - 1}$
    The result of the chain rule is:
    
    $\frac{4^{x} \log{\left(4 \right)} \cos{\left(\log{\left(4^{x} - 1 \right)} \right)}}{4^{x} - 1}$
  Now plug in to the quotient rule:
  
  $\frac{- \frac{4^{x} \log{\left(4 \right)} \sin^{2}{\left(\log{\left(4^{x} - 1 \right)} \right)}}{4^{x} - 1} - \frac{4^{x} \log{\left(4 \right)} \cos^{2}{\left(\log{\left(4^{x} - 1 \right)} \right)}}{4^{x} - 1}}{\sin^{2}{\left(\log{\left(4^{x} - 1 \right)} \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 x /        2/   / x    \\\       
4 *\-1 - cot \log\4  - 1///*log(4)
----------------------------------
               x                  
              4  - 1

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

Simplify

The second derivative [src]

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

$$\frac{4^{x} \left(\cot^{2}{\left(\log{\left(4^{x} - 1 \right)} \right)} + 1\right) \left(\frac{2 \cdot 4^{x} \cot{\left(\log{\left(4^{x} - 1 \right)} \right)}}{4^{x} - 1} + \frac{4^{x}}{4^{x} - 1} - 1\right) \log{\left(4 \right)}^{2}}{4^{x} - 1}$$

Simplify

The third derivative [src]

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

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

Simplify

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Similar expressions

Derivative of ctg(ln(4^x-1))

The graph:

Piecewise:

The solution

Method #1

Method #2