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Derivative of:
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Identical expressions
y=ln*((e^x-e^-x)/ two)
y equally ln multiply by ((e to the power of x minus e to the power of minus x) divide by 2)
y equally ln multiply by ((e to the power of x minus e to the power of minus x) divide by two)
y=ln*((ex-e-x)/2)
y=ln*ex-e-x/2
y=ln((e^x-e^-x)/2)
y=ln((ex-e-x)/2)
y=lnex-e-x/2
y=lne^x-e^-x/2
y=ln*((e^x-e^-x) divide by 2)
Similar expressions
y=ln*((e^x+e^-x)/2)
y=ln*((e^x-e^+x)/2)

The derivative of the function/ y=ln*((e^x-e^-x)/2)

Derivative of y=ln*((e^x-e^-x)/2)

The solution

You have entered [src]

   / x    -x\
   |E  - E  |
log|--------|
   \   2    /

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

log((E^x - E^(-x))/2)

Detail solution

Let $u = \frac{e^{x} - e^{- x}}{2}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{e^{x} - e^{- x}}{2}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Differentiate $e^{x} - e^{- x}$ term by term:
    1. The derivative of $e^{x}$ is itself.
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Let $u = - x$ .
      2. The derivative of $e^{u}$ is itself.
      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(- x\right)$ :
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $-1$
        
        The result of the chain rule is:
        
        $- e^{- x}$
      So, the result is: $e^{- x}$
    The result is: $e^{x} + e^{- x}$
  So, the result is: $\frac{e^{x}}{2} + \frac{e^{- x}}{2}$
The result of the chain rule is:

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

$\frac{1}{\tanh{\left(x \right)}}$

The answer is:

$\frac{1}{\tanh{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  / x    -x\
  |e    e  |
2*|-- + ---|
  \2     2 /
------------
   x    -x  
  E  - E

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of y=ln*((e^x-e^-x)/2)

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