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Identical expressions
tanx/(x^ two - four)
tangent of x divide by (x squared minus 4)
tangent of x divide by (x to the power of two minus four)
tanx/(x2-4)
tanx/x2-4
tanx/(x²-4)
tanx/(x to the power of 2-4)
tanx/x^2-4
tanx divide by (x^2-4)
Similar expressions
tanx/(x^2+4)

The derivative of the function/ tanx/(x^2-4)

Derivative of tanx/(x^2-4)

The solution

You have entered [src]

tan(x)
------
 2    
x  - 4

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

tan(x)/(x^2 - 4)

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 $g{\left(x \right)} = x^{2} - 4$ .

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. Differentiate $x^{2} - 4$ term by term:
  1. The derivative of the constant $-4$ is zero.
  2. Apply the power rule: $x^{2}$ goes to $2 x$
  The result is: $2 x$
Now plug in to the quotient rule:

$\frac{- 2 x \tan{\left(x \right)} + \frac{\left(x^{2} - 4\right) \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}}}{\left(x^{2} - 4\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2                
1 + tan (x)   2*x*tan(x)
----------- - ----------
    2                 2 
   x  - 4     / 2    \  
              \x  - 4/

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

Simplify

The second derivative [src]

  /                       /          2 \                           \
  |                       |       4*x  |                           |
  |                       |-1 + -------|*tan(x)                    |
  |                       |           2|              /       2   \|
  |/       2   \          \     -4 + x /          2*x*\1 + tan (x)/|
2*|\1 + tan (x)/*tan(x) + --------------------- - -----------------|
  |                                    2                     2     |
  \                              -4 + x                -4 + x      /
--------------------------------------------------------------------
                                    2                               
                              -4 + x

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of tanx/(x^2-4)

The graph:

Piecewise:

The solution