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tgx/(8x^ two + five)
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tgx/(8x2+5)
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Similar expressions
tgx/(8x^2-5)

The derivative of the function/ tgx/(8x^2+5)

Derivative of tgx/(8x^2+5)

The solution

You have entered [src]

 tan(x) 
--------
   2    
8*x  + 5

$$\frac{\tan{\left(x \right)}}{8 x^{2} + 5}$$

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 $g{\left(x \right)} = 8 x^{2} + 5$ .

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 $8 x^{2} + 5$ term by term:
  1. The derivative of the constant $5$ is zero.
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    So, the result is: $16 x$
  The result is: $16 x$
Now plug in to the quotient rule:

$\frac{- 16 x \tan{\left(x \right)} + \frac{\left(8 x^{2} + 5\right) \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}}}{\left(8 x^{2} + 5\right)^{2}}$
Now simplify:

$\frac{8 x^{2} - 8 x \sin{\left(2 x \right)} + 5}{\left(8 x^{2} + 5\right)^{2} \cos^{2}{\left(x \right)}}$

The answer is:

$\frac{8 x^{2} - 8 x \sin{\left(2 x \right)} + 5}{\left(8 x^{2} + 5\right)^{2} \cos^{2}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2                 
1 + tan (x)   16*x*tan(x)
----------- - -----------
     2                  2
  8*x  + 5    /   2    \ 
              \8*x  + 5/

$$- \frac{16 x \tan{\left(x \right)}}{\left(8 x^{2} + 5\right)^{2}} + \frac{\tan^{2}{\left(x \right)} + 1}{8 x^{2} + 5}$$

Simplify

The second derivative [src]

  /                                              /          2  \       \
  |                                              |      32*x   |       |
  |                                            8*|-1 + --------|*tan(x)|
  |                            /       2   \     |            2|       |
  |/       2   \          16*x*\1 + tan (x)/     \     5 + 8*x /       |
2*|\1 + tan (x)/*tan(x) - ------------------ + ------------------------|
  |                                   2                       2        |
  \                            5 + 8*x                 5 + 8*x         /
------------------------------------------------------------------------
                                       2                                
                                5 + 8*x

$$\frac{2 \left(- \frac{16 x \left(\tan^{2}{\left(x \right)} + 1\right)}{8 x^{2} + 5} + \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \frac{8 \left(\frac{32 x^{2}}{8 x^{2} + 5} - 1\right) \tan{\left(x \right)}}{8 x^{2} + 5}\right)}{8 x^{2} + 5}$$

Simplify

The third derivative [src]

  /                                                 /          2  \         /          2  \                                   \
  |                                   /       2   \ |      32*x   |         |      16*x   |                                   |
  |                                24*\1 + tan (x)/*|-1 + --------|   768*x*|-1 + --------|*tan(x)                            |
  |                                                 |            2|         |            2|               /       2   \       |
  |/       2   \ /         2   \                    \     5 + 8*x /         \     5 + 8*x /          48*x*\1 + tan (x)/*tan(x)|
2*|\1 + tan (x)/*\1 + 3*tan (x)/ + -------------------------------- - ---------------------------- - -------------------------|
  |                                                   2                                 2                            2        |
  |                                            5 + 8*x                        /       2\                      5 + 8*x         |
  \                                                                           \5 + 8*x /                                      /
-------------------------------------------------------------------------------------------------------------------------------
                                                                   2                                                           
                                                            5 + 8*x

$$\frac{2 \left(- \frac{48 x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{8 x^{2} + 5} - \frac{768 x \left(\frac{16 x^{2}}{8 x^{2} + 5} - 1\right) \tan{\left(x \right)}}{\left(8 x^{2} + 5\right)^{2}} + \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) + \frac{24 \left(\frac{32 x^{2}}{8 x^{2} + 5} - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{8 x^{2} + 5}\right)}{8 x^{2} + 5}$$

Simplify

The graph

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Derivative of tgx/(8x^2+5)

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Piecewise:

The solution