Derivative of y=lntg^2x

The solution

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   2        
log (tan(x))

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

log(tan(x))^2

Detail solution

Let $u = \log{\left(\tan{\left(x \right)} \right)}$ .
Apply the power rule: $u^{2}$ goes to $2 u$
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:

$\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \log{\left(\tan{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)} \tan{\left(x \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of y=lntg^2x

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The solution