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Identical expressions
(exp(2x)- one)/(exp(2x)+ one)
( exponent of (2x) minus 1) divide by ( exponent of (2x) plus 1)
( exponent of (2x) minus one) divide by ( exponent of (2x) plus one)
exp2x-1/exp2x+1
(exp(2x)-1) divide by (exp(2x)+1)
Similar expressions
(exp(2x)-1)/(exp(2x)-1)
(exp(2x)+1)/(exp(2x)+1)

The derivative of the function/ (exp(2x)-1)/(exp(2x)+1)

Derivative of (exp(2x)-1)/(exp(2x)+1)

The solution

You have entered [src]

 2*x    
e    - 1
--------
 2*x    
e    + 1

$$\frac{e^{2 x} - 1}{e^{2 x} + 1}$$

(exp(2*x) - 1)/(exp(2*x) + 1)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = e^{2 x} - 1$ and $g{\left(x \right)} = e^{2 x} + 1$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $e^{2 x} - 1$ term by term:
  1. The derivative of the constant $-1$ is zero.
  2. Let $u = 2 x$ .
  3. The derivative of $e^{u}$ is itself.
  4. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
    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: $2$
    The result of the chain rule is:
    
    $2 e^{2 x}$
  The result is: $2 e^{2 x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $e^{2 x} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Let $u = 2 x$ .
  3. The derivative of $e^{u}$ is itself.
  4. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
    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: $2$
    The result of the chain rule is:
    
    $2 e^{2 x}$
  The result is: $2 e^{2 x}$
Now plug in to the quotient rule:

$\frac{- 2 \left(e^{2 x} - 1\right) e^{2 x} + 2 \left(e^{2 x} + 1\right) e^{2 x}}{\left(e^{2 x} + 1\right)^{2}}$
Now simplify:

$\frac{1}{\cosh^{2}{\left(x \right)}}$

The answer is:

$\frac{1}{\cosh^{2}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    2*x      / 2*x    \  2*x
 2*e       2*\e    - 1/*e   
-------- - -----------------
 2*x                    2   
e    + 1      / 2*x    \    
              \e    + 1/

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

Simplify

The second derivative [src]

  /               /        2*x \            \     
  |               |     2*e    | /      2*x\|     
  |               |1 - --------|*\-1 + e   /|     
  |        2*x    |         2*x|            |     
  |     2*e       \    1 + e   /            |  2*x
4*|1 - -------- - --------------------------|*e   
  |         2*x                 2*x         |     
  \    1 + e               1 + e            /     
--------------------------------------------------
                          2*x                     
                     1 + e

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

Simplify

The third derivative [src]

  /                           /        2*x          4*x  \                        \     
  |               /      2*x\ |     6*e          6*e     |     /        2*x \     |     
  |               \-1 + e   /*|1 - -------- + -----------|     |     2*e    |  2*x|     
  |                           |         2*x             2|   3*|1 - --------|*e   |     
  |        2*x                |    1 + e      /     2*x\ |     |         2*x|     |     
  |     3*e                   \               \1 + e   / /     \    1 + e   /     |  2*x
8*|1 - -------- - ---------------------------------------- - ---------------------|*e   
  |         2*x                        2*x                               2*x      |     
  \    1 + e                      1 + e                             1 + e         /     
----------------------------------------------------------------------------------------
                                             2*x                                        
                                        1 + e

$$\frac{8 \left(- \frac{3 \left(1 - \frac{2 e^{2 x}}{e^{2 x} + 1}\right) e^{2 x}}{e^{2 x} + 1} - \frac{\left(e^{2 x} - 1\right) \left(1 - \frac{6 e^{2 x}}{e^{2 x} + 1} + \frac{6 e^{4 x}}{\left(e^{2 x} + 1\right)^{2}}\right)}{e^{2 x} + 1} + 1 - \frac{3 e^{2 x}}{e^{2 x} + 1}\right) e^{2 x}}{e^{2 x} + 1}$$

Simplify

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Derivative of (exp(2x)-1)/(exp(2x)+1)

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Piecewise:

The solution