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Identical expressions
tan(x)/((three *x))
tangent of (x) divide by ((3 multiply by x))
tangent of (x) divide by ((three multiply by x))
tan(x)/((3x))
tanx/3x
tan(x) divide by ((3*x))

The derivative of the function/ tan(x)/((3*x))

Derivative of tan(x)/((3*x))

The solution

You have entered [src]

tan(x)
------
 3*x

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

tan(x)/((3*x))

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 $g{\left(x \right)} = 3 x$ .

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. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $x$ goes to $1$
  So, the result is: $3$
Now plug in to the quotient rule:

$\frac{\frac{3 x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} - 3 \tan{\left(x \right)}}{9 x^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Identical expressions

Derivative of tan(x)/((3*x))

The graph:

Piecewise:

The solution