Derivative of y=tgx/x^3

The solution

You have entered [src]

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

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

tan(x)/x^3

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

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: $x^{3}$ goes to $3 x^{2}$
Now plug in to the quotient rule:

$\frac{\frac{x^{3} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} - 3 x^{2} \tan{\left(x \right)}}{x^{6}}$
Now simplify:

$\frac{x - \frac{3 \sin{\left(2 x \right)}}{2}}{x^{4} \cos^{2}{\left(x \right)}}$

The answer is:

$\frac{x - \frac{3 \sin{\left(2 x \right)}}{2}}{x^{4} \cos^{2}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2              
1 + tan (x)   3*tan(x)
----------- - --------
      3           4   
     x           x

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                                               /       2   \     /       2   \       \
  |/       2   \ /         2   \   30*tan(x)   18*\1 + tan (x)/   9*\1 + tan (x)/*tan(x)|
2*|\1 + tan (x)/*\1 + 3*tan (x)/ - --------- + ---------------- - ----------------------|
  |                                     3              2                    x           |
  \                                    x              x                                 /
-----------------------------------------------------------------------------------------
                                             3                                           
                                            x

$$\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) - \frac{9 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{x} + \frac{18 \left(\tan^{2}{\left(x \right)} + 1\right)}{x^{2}} - \frac{30 \tan{\left(x \right)}}{x^{3}}\right)}{x^{3}}$$

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Derivative of y=tgx/x^3

The graph:

Piecewise:

The solution