Derivative of tg(acos(tg(x)))

The solution

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tan(acos(tan(x)))

$$\tan{\left(\operatorname{acos}{\left(\tan{\left(x \right)} \right)} \right)}$$

tan(acos(tan(x)))

Detail solution

Rewrite the function to be differentiated:

$\tan{\left(\operatorname{acos}{\left(\tan{\left(x \right)} \right)} \right)} = \frac{\sqrt{1 - \tan^{2}{\left(x \right)}}}{\tan{\left(x \right)}}$
Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sqrt{1 - \tan^{2}{\left(x \right)}}$ and $g{\left(x \right)} = \tan{\left(x \right)}$ .

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

$\frac{- \frac{\sqrt{1 - \tan^{2}{\left(x \right)}} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} - \frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\sqrt{1 - \tan^{2}{\left(x \right)}} \cos^{2}{\left(x \right)}}}{\tan^{2}{\left(x \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 /       2   \ /       2              \ 
-\1 + tan (x)/*\1 + tan (acos(tan(x)))/ 
----------------------------------------
               _____________            
              /        2                
            \/  1 - tan (x)

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

Simplify

The second derivative [src]

                                        /                   /       2   \            /       2   \                  \
 /       2   \ /       2              \ |    2*tan(x)       \1 + tan (x)/*tan(x)   2*\1 + tan (x)/*tan(acos(tan(x)))|
-\1 + tan (x)/*\1 + tan (acos(tan(x)))/*|---------------- + -------------------- + ---------------------------------|
                                        |   _____________                  3/2                        2             |
                                        |  /        2         /       2   \                   -1 + tan (x)          |
                                        \\/  1 - tan (x)      \1 - tan (x)/                                         /

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

Simplify

The third derivative [src]

                                       /                2                                                                                   2                                     2                          2                                                                                       2                         \
                                       |   /       2   \             2            /       2   \         2    /       2   \     /       2   \     2                   /       2   \     2        /       2   \  /       2              \      /       2   \                              /       2   \                          |
/       2   \ /       2              \ |   \1 + tan (x)/        4*tan (x)       2*\1 + tan (x)/    6*tan (x)*\1 + tan (x)/   4*\1 + tan (x)/ *tan (acos(tan(x)))   3*\1 + tan (x)/ *tan (x)   2*\1 + tan (x)/ *\1 + tan (acos(tan(x)))/   12*\1 + tan (x)/*tan(x)*tan(acos(tan(x)))   6*\1 + tan (x)/ *tan(x)*tan(acos(tan(x)))|
\1 + tan (x)/*\1 + tan (acos(tan(x)))/*|- ---------------- - ---------------- - ---------------- - ----------------------- - ----------------------------------- - ------------------------ - ----------------------------------------- - ----------------------------------------- + -----------------------------------------|
                                       |               3/2      _____________      _____________                    3/2                             3/2                             5/2                                 3/2                                      2                                               2             |
                                       |  /       2   \        /        2         /        2           /       2   \                   /       2   \                   /       2   \                       /       2   \                                 -1 + tan (x)                              /        2   \              |
                                       \  \1 - tan (x)/      \/  1 - tan (x)    \/  1 - tan (x)        \1 - tan (x)/                   \1 - tan (x)/                   \1 - tan (x)/                       \1 - tan (x)/                                                                           \-1 + tan (x)/              /

$$\left(\tan^{2}{\left(x \right)} + 1\right) \left(\tan^{2}{\left(\operatorname{acos}{\left(\tan{\left(x \right)} \right)} \right)} + 1\right) \left(- \frac{12 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} \tan{\left(\operatorname{acos}{\left(\tan{\left(x \right)} \right)} \right)}}{\tan^{2}{\left(x \right)} - 1} + \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \tan{\left(x \right)} \tan{\left(\operatorname{acos}{\left(\tan{\left(x \right)} \right)} \right)}}{\left(\tan^{2}{\left(x \right)} - 1\right)^{2}} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{\sqrt{1 - \tan^{2}{\left(x \right)}}} - \frac{4 \tan^{2}{\left(x \right)}}{\sqrt{1 - \tan^{2}{\left(x \right)}}} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \left(\tan^{2}{\left(\operatorname{acos}{\left(\tan{\left(x \right)} \right)} \right)} + 1\right)}{\left(1 - \tan^{2}{\left(x \right)}\right)^{\frac{3}{2}}} - \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \tan^{2}{\left(\operatorname{acos}{\left(\tan{\left(x \right)} \right)} \right)}}{\left(1 - \tan^{2}{\left(x \right)}\right)^{\frac{3}{2}}} - \frac{\left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\left(1 - \tan^{2}{\left(x \right)}\right)^{\frac{3}{2}}} - \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)}}{\left(1 - \tan^{2}{\left(x \right)}\right)^{\frac{3}{2}}} - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \tan^{2}{\left(x \right)}}{\left(1 - \tan^{2}{\left(x \right)}\right)^{\frac{5}{2}}}\right)$$

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Derivative of tg(acos(tg(x)))

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