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Identical expressions
three ^(two *x)*cot(log(x))
3 to the power of (2 multiply by x) multiply by cotangent of ( logarithm of (x))
three to the power of (two multiply by x) multiply by cotangent of ( logarithm of (x))
3(2*x)*cot(log(x))
32*x*cotlogx
3^(2x)cot(log(x))
3(2x)cot(log(x))
32xcotlogx
3^2xcotlogx

The derivative of the function/ 3^(2*x)*cot(log(x))

Derivative of 3^(2x)cot(log(x))

The solution

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 2*x            
3   *cot(log(x))

3^{2 x} \cot{\left(\log{\left(x \right)} \right)}

d / 2*x            \
--\3   *cot(log(x))/
dx

\frac{d}{d x} 3^{2 x} \cot{\left(\log{\left(x \right)} \right)}

Transform

Detail solution

Apply the product rule:

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

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

$\frac{9^{x} \left(2 x \log{\left(3 \right)} \sin{\left(2 \log{\left(x \right)} \right)} - 2\right)}{x \left(1 - \cos{\left(2 \log{\left(x \right)} \right)}\right)}$

The answer is:

\frac{9^{x} \left(2 x \log{\left(3 \right)} \sin{\left(2 \log{\left(x \right)} \right)} - 2\right)}{x \left(1 - \cos{\left(2 \log{\left(x \right)} \right)}\right)}

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Derivative of 3^(2x)cot(log(x))

The graph:

Piecewise:

The solution

Method #1

Method #2

Other calculators

How to use it?

Identical expressions

Derivative of 3^(2*x)*cot(log(x))

The graph:

Piecewise:

The solution

Method #1

Method #2

Derivative of 3^(2x)cot(log(x))