Derivative of tgx/x

The solution

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tan(x)
------
  x

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

d /tan(x)\
--|------|
dx\  x   /

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

Transform

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \tan{\left(x \right)}$ and $g{\left(x \right)} = x$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = \cos{\left(x \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  Now plug in to the quotient rule:
  
  $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2            
1 + tan (x)   tan(x)
----------- - ------
     x           2  
                x

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of tgx/x

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The solution