Derivative of tg(ln^3x)

The solution

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   /   3   \
tan\log (x)/

$$\tan{\left(\log{\left(x \right)}^{3} \right)}$$

d /   /   3   \\
--\tan\log (x)//
dx

$$\frac{d}{d x} \tan{\left(\log{\left(x \right)}^{3} \right)}$$

Transform

Detail solution

Rewrite the function to be differentiated:

$\tan{\left(\log{\left(x \right)}^{3} \right)} = \frac{\sin{\left(\log{\left(x \right)}^{3} \right)}}{\cos{\left(\log{\left(x \right)}^{3} \right)}}$
Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \sin{\left(\log{\left(x \right)}^{3} \right)}$ and $g{\left(x \right)} = \cos{\left(\log{\left(x \right)}^{3} \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \log{\left(x \right)}^{3}$ .
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 $\frac{d}{d x} \log{\left(x \right)}^{3}$ :
  1. Let $u = \log{\left(x \right)}$ .
  2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
  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:
    
    $\frac{3 \log{\left(x \right)}^{2}}{x}$
  The result of the chain rule is:
  
  $\frac{3 \log{\left(x \right)}^{2} \cos{\left(\log{\left(x \right)}^{3} \right)}}{x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \log{\left(x \right)}^{3}$ .
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 $\frac{d}{d x} \log{\left(x \right)}^{3}$ :
  1. Let $u = \log{\left(x \right)}$ .
  2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
  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:
    
    $\frac{3 \log{\left(x \right)}^{2}}{x}$
  The result of the chain rule is:
  
  $- \frac{3 \log{\left(x \right)}^{2} \sin{\left(\log{\left(x \right)}^{3} \right)}}{x}$
Now plug in to the quotient rule:

$\frac{\frac{3 \log{\left(x \right)}^{2} \sin^{2}{\left(\log{\left(x \right)}^{3} \right)}}{x} + \frac{3 \log{\left(x \right)}^{2} \cos^{2}{\left(\log{\left(x \right)}^{3} \right)}}{x}}{\cos^{2}{\left(\log{\left(x \right)}^{3} \right)}}$
Now simplify:

$\frac{3 \log{\left(x \right)}^{2}}{x \cos^{2}{\left(\log{\left(x \right)}^{3} \right)}}$

The answer is:

$\frac{3 \log{\left(x \right)}^{2}}{x \cos^{2}{\left(\log{\left(x \right)}^{3} \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /       2/   3   \\ /       2                      4       /   3   \        6    /       2/   3   \\         3       /   3   \         6       2/   3   \\
6*\1 + tan \log (x)//*\1 + log (x) - 3*log(x) - 9*log (x)*tan\log (x)/ + 9*log (x)*\1 + tan \log (x)// + 18*log (x)*tan\log (x)/ + 18*log (x)*tan \log (x)//
------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                              3                                                                             
                                                                             x

$$\frac{6 \left(\tan^{2}{\left(\log{\left(x \right)}^{3} \right)} + 1\right) \left(18 \log{\left(x \right)}^{6} \tan^{2}{\left(\log{\left(x \right)}^{3} \right)} + 9 \left(\tan^{2}{\left(\log{\left(x \right)}^{3} \right)} + 1\right) \log{\left(x \right)}^{6} - 9 \log{\left(x \right)}^{4} \tan{\left(\log{\left(x \right)}^{3} \right)} + 18 \log{\left(x \right)}^{3} \tan{\left(\log{\left(x \right)}^{3} \right)} + \log{\left(x \right)}^{2} - 3 \log{\left(x \right)} + 1\right)}{x^{3}}$$

Simplify

The graph

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Derivative of tg(ln^3x)

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The solution