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Derivative of:
Derivative of x^3/cosx
Derivative of (x-6)x^3
Derivative of e^(3/x)
Derivative of y=x(x-1)
Identical expressions
tg^ two (sqrt(lnx))
tg squared ( square root of (lnx))
tg to the power of two ( square root of (lnx))
tg^2(√(lnx))
tg2(sqrt(lnx))
tg2sqrtlnx
tg²(sqrt(lnx))
tg to the power of 2(sqrt(lnx))
tg^2sqrtlnx

The derivative of the function/ tg^2(sqrt(lnx))

Derivative of tg^2(sqrt(lnx))

The solution

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   2/  ________\
tan \\/ log(x) /

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

tan(sqrt(log(x)))^2

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                       /   3/  ________\        2/  ________\        /  ________\     /       2/  ________\\        2/  ________\     /       2/  ________\\        /  ________\        /  ________\     /       2/  ________\\    /  ________\\
/       2/  ________\\ |tan \\/ log(x) /   3*tan \\/ log(x) /   2*tan\\/ log(x) /   3*\1 + tan \\/ log(x) //   3*tan \\/ log(x) /   3*\1 + tan \\/ log(x) //   3*tan\\/ log(x) /   3*tan\\/ log(x) /   2*\1 + tan \\/ log(x) //*tan\\/ log(x) /|
\1 + tan \\/ log(x) //*|---------------- - ------------------ + ----------------- - ------------------------ - ------------------ - ------------------------ + ----------------- + ----------------- + ----------------------------------------|
                       |      3/2                log(x)               ________              2*log(x)                    2                       2                      3/2                 5/2                           3/2                   |
                       \   log   (x)                                \/ log(x)                                      2*log (x)               4*log (x)              2*log   (x)         4*log   (x)                     log   (x)                /
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                        3                                                                                                                       
                                                                                                                       x

$$\frac{\left(\tan^{2}{\left(\sqrt{\log{\left(x \right)}} \right)} + 1\right) \left(- \frac{3 \left(\tan^{2}{\left(\sqrt{\log{\left(x \right)}} \right)} + 1\right)}{2 \log{\left(x \right)}} - \frac{3 \left(\tan^{2}{\left(\sqrt{\log{\left(x \right)}} \right)} + 1\right)}{4 \log{\left(x \right)}^{2}} + \frac{2 \left(\tan^{2}{\left(\sqrt{\log{\left(x \right)}} \right)} + 1\right) \tan{\left(\sqrt{\log{\left(x \right)}} \right)}}{\log{\left(x \right)}^{\frac{3}{2}}} - \frac{3 \tan^{2}{\left(\sqrt{\log{\left(x \right)}} \right)}}{\log{\left(x \right)}} - \frac{3 \tan^{2}{\left(\sqrt{\log{\left(x \right)}} \right)}}{2 \log{\left(x \right)}^{2}} + \frac{2 \tan{\left(\sqrt{\log{\left(x \right)}} \right)}}{\sqrt{\log{\left(x \right)}}} + \frac{\tan^{3}{\left(\sqrt{\log{\left(x \right)}} \right)}}{\log{\left(x \right)}^{\frac{3}{2}}} + \frac{3 \tan{\left(\sqrt{\log{\left(x \right)}} \right)}}{2 \log{\left(x \right)}^{\frac{3}{2}}} + \frac{3 \tan{\left(\sqrt{\log{\left(x \right)}} \right)}}{4 \log{\left(x \right)}^{\frac{5}{2}}}\right)}{x^{3}}$$

Simplify

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Derivative of tg^2(sqrt(lnx))

The graph:

Piecewise:

The solution