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Derivative of:
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Derivative of atan(log(x))+log(atan(x))
Identical expressions
y=ln^ two *(tg10*x)
y equally ln squared multiply by (tg10 multiply by x)
y equally ln to the power of two multiply by (tg10 multiply by x)
y=ln2*(tg10*x)
y=ln2*tg10*x
y=ln²*(tg10*x)
y=ln to the power of 2*(tg10*x)
y=ln^2(tg10x)
y=ln2(tg10x)
y=ln2tg10x
y=ln^2tg10x

The derivative of the function/ y=ln^2*(tg10*x)

Derivative of y=ln^2(tg10x)

The solution

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   2           
log (tan(10*x))

$$\log{\left(\tan{\left(10 x \right)} \right)}^{2}$$

log(tan(10*x))^2

Detail solution

Let $u = \log{\left(\tan{\left(10 x \right)} \right)}$ .
Apply the power rule: $u^{2}$ goes to $2 u$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(\tan{\left(10 x \right)} \right)}$ :
1. Let $u = \tan{\left(10 x \right)}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(10 x \right)}$ :
  1. Rewrite the function to be differentiated:
    
    $\tan{\left(10 x \right)} = \frac{\sin{\left(10 x \right)}}{\cos{\left(10 x \right)}}$
  2. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \sin{\left(10 x \right)}$ and $g{\left(x \right)} = \cos{\left(10 x \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Let $u = 10 x$ .
    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 $\frac{d}{d x} 10 x$ :
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $10$
      The result of the chain rule is:
      
      $10 \cos{\left(10 x \right)}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Let $u = 10 x$ .
    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 $\frac{d}{d x} 10 x$ :
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $10$
      The result of the chain rule is:
      
      $- 10 \sin{\left(10 x \right)}$
    Now plug in to the quotient rule:
    
    $\frac{10 \sin^{2}{\left(10 x \right)} + 10 \cos^{2}{\left(10 x \right)}}{\cos^{2}{\left(10 x \right)}}$
  The result of the chain rule is:
  
  $\frac{10 \sin^{2}{\left(10 x \right)} + 10 \cos^{2}{\left(10 x \right)}}{\cos^{2}{\left(10 x \right)} \tan{\left(10 x \right)}}$
The result of the chain rule is:

$\frac{2 \left(10 \sin^{2}{\left(10 x \right)} + 10 \cos^{2}{\left(10 x \right)}\right) \log{\left(\tan{\left(10 x \right)} \right)}}{\cos^{2}{\left(10 x \right)} \tan{\left(10 x \right)}}$
Now simplify:

$\frac{40 \log{\left(\tan{\left(10 x \right)} \right)}}{\sin{\left(20 x \right)}}$

The answer is:

$\frac{40 \log{\left(\tan{\left(10 x \right)} \right)}}{\sin{\left(20 x \right)}}$

The graph

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Plot the graph f'(x)

The first derivative [src]

  /           2      \               
2*\10 + 10*tan (10*x)/*log(tan(10*x))
-------------------------------------
              tan(10*x)

$$\frac{2 \left(10 \tan^{2}{\left(10 x \right)} + 10\right) \log{\left(\tan{\left(10 x \right)} \right)}}{\tan{\left(10 x \right)}}$$

Simplify

The second derivative [src]

                     /                          2         /       2      \               \
    /       2      \ |                   1 + tan (10*x)   \1 + tan (10*x)/*log(tan(10*x))|
200*\1 + tan (10*x)/*|2*log(tan(10*x)) + -------------- - -------------------------------|
                     |                        2                         2                |
                     \                     tan (10*x)                tan (10*x)          /

$$200 \left(\tan^{2}{\left(10 x \right)} + 1\right) \left(- \frac{\left(\tan^{2}{\left(10 x \right)} + 1\right) \log{\left(\tan{\left(10 x \right)} \right)}}{\tan^{2}{\left(10 x \right)}} + \frac{\tan^{2}{\left(10 x \right)} + 1}{\tan^{2}{\left(10 x \right)}} + 2 \log{\left(\tan{\left(10 x \right)} \right)}\right)$$

Simplify

The third derivative [src]

                      /                    2                                                                                                           2               \
                      |    /       2      \                                   /       2      \     /       2      \                    /       2      \                |
     /       2      \ |  3*\1 + tan (10*x)/                                 6*\1 + tan (10*x)/   4*\1 + tan (10*x)/*log(tan(10*x))   2*\1 + tan (10*x)/ *log(tan(10*x))|
2000*\1 + tan (10*x)/*|- ------------------- + 4*log(tan(10*x))*tan(10*x) + ------------------ - --------------------------------- + ----------------------------------|
                      |          3                                              tan(10*x)                    tan(10*x)                              3                  |
                      \       tan (10*x)                                                                                                         tan (10*x)            /

$$2000 \left(\tan^{2}{\left(10 x \right)} + 1\right) \left(\frac{2 \left(\tan^{2}{\left(10 x \right)} + 1\right)^{2} \log{\left(\tan{\left(10 x \right)} \right)}}{\tan^{3}{\left(10 x \right)}} - \frac{3 \left(\tan^{2}{\left(10 x \right)} + 1\right)^{2}}{\tan^{3}{\left(10 x \right)}} - \frac{4 \left(\tan^{2}{\left(10 x \right)} + 1\right) \log{\left(\tan{\left(10 x \right)} \right)}}{\tan{\left(10 x \right)}} + \frac{6 \left(\tan^{2}{\left(10 x \right)} + 1\right)}{\tan{\left(10 x \right)}} + 4 \log{\left(\tan{\left(10 x \right)} \right)} \tan{\left(10 x \right)}\right)$$

Simplify

The graph

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Identical expressions

Derivative of y=ln^2(tg10x)

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Piecewise:

The solution

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Identical expressions

Derivative of y=ln^2*(tg10*x)

The graph:

Piecewise:

The solution

Derivative of y=ln^2(tg10x)