Derivative of tan(1/x)

The solution

You have entered [src]

   /  1\
tan|1*-|
   \  x/

$$\tan{\left(1 \cdot \frac{1}{x} \right)}$$

d /   /  1\\
--|tan|1*-||
dx\   \  x//

$$\frac{d}{d x} \tan{\left(1 \cdot \frac{1}{x} \right)}$$

Transform

Detail solution

Rewrite the function to be differentiated:

$\tan{\left(1 \cdot \frac{1}{x} \right)} = \frac{\sin{\left(1 \cdot \frac{1}{x} \right)}}{\cos{\left(1 \cdot \frac{1}{x} \right)}}$
Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sin{\left(1 \cdot \frac{1}{x} \right)}$ and $g{\left(x \right)} = \cos{\left(1 \cdot \frac{1}{x} \right)}$ .

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

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

$- \frac{1}{x^{2} \cos^{2}{\left(\frac{1}{x} \right)}}$

The answer is:

$- \frac{1}{x^{2} \cos^{2}{\left(\frac{1}{x} \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 /       2/  1\\ 
-|1 + tan |1*-|| 
 \        \  x// 
-----------------
         2       
        x

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

Simplify

The second derivative [src]

                /       /1\\
                |    tan|-||
  /       2/1\\ |       \x/|
2*|1 + tan |-||*|1 + ------|
  \        \x// \      x   /
----------------------------
              3             
             x

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

Simplify

The third derivative [src]

                 /           2/1\        2/1\        /1\\
                 |    1 + tan |-|   2*tan |-|   6*tan|-||
   /       2/1\\ |            \x/         \x/        \x/|
-2*|1 + tan |-||*|3 + ----------- + --------- + --------|
   \        \x// |          2            2         x    |
                 \         x            x               /
---------------------------------------------------------
                             4                           
                            x

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

Simplify

The graph

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Derivative of tan(1/x)

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The solution