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log(tan)^(two)*(x)
logarithm of ( tangent of ) to the power of (2) multiply by (x)
logarithm of ( tangent of ) to the power of (two) multiply by (x)
log(tan)(2)*(x)
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log(tan)^(2)(x)
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logtan2x
logtan^2x

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

Derivative of log(tan)^(2)*(x)

The solution

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   2          
log (tan(x))*x

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

log(tan(x))^2*x

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of log(tan)^(2)*(x)

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