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Derivative of:
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Derivative of e^sin(x)^(2)
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Identical expressions
ln(ctg^ two)*x
ln(ctg squared ) multiply by x
ln(ctg to the power of two) multiply by x
ln(ctg2)*x
lnctg2*x
ln(ctg²)*x
ln(ctg to the power of 2)*x
ln(ctg^2)x
ln(ctg2)x
lnctg2x
lnctg^2x

The derivative of the function/ ln(ctg^2)*x

Derivative of ln(ctg^2)*x

The solution

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   /   2   \  
log\cot (x)/*x

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

log(cot(x)^2)*x

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Identical expressions

Derivative of ln(ctg^2)*x

The graph:

Piecewise:

The solution

Method #1

Method #2