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(e^(2x)-1)/(6e^x)

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Identical expressions
(e^(2x)- one)/(6e^x)
(e to the power of (2x) minus 1) divide by (6e to the power of x)
(e to the power of (2x) minus one) divide by (6e to the power of x)
(e(2x)-1)/(6ex)
e2x-1/6ex
e^2x-1/6e^x
(e^(2x)-1) divide by (6e^x)
Similar expressions
(e^(2x)+1)/(6e^x)

The derivative of the function/ (e^(2x)-1)/(6e^x)

Derivative of (e^(2x)-1)/(6e^x)

The solution

You have entered [src]

 2*x    
E    - 1
--------
     x  
  6*E

$$\frac{e^{2 x} - 1}{6 e^{x}}$$

(E^(2*x) - 1)/((6*E^x))

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 $g{\left(x \right)} = 6 e^{x}$ .

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. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of $e^{x}$ is itself.
  So, the result is: $6 e^{x}$
Now plug in to the quotient rule:

$\frac{\left(- 6 \left(e^{2 x} - 1\right) e^{x} + 12 e^{3 x}\right) e^{- 2 x}}{36}$
Now simplify:

$\frac{\cosh{\left(x \right)}}{3}$

The answer is:

$\frac{\cosh{\left(x \right)}}{3}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   -x        / 2*x    \  -x
  e    2*x   \E    - 1/*e  
2*---*e    - --------------
   6               6

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

Simplify

The second derivative [src]

/      2*x\  -x
\-1 + e   /*e  
---------------
       6

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

Simplify

The third derivative [src]

   x   /      2*x\  -x
2*e  - \-1 + e   /*e  
----------------------
          6

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

Simplify

The graph

Derivative of (e^(2x)-1)/(6e^x)