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Derivative of:
Derivative of x^(3/2)-3*x+1
Derivative of x^3/(2(x+1)^2)
Derivative of (x-2)/(x+3)
Derivative of x^2+2/x
Integral of d{x}:
e^(2*x)/x
Limit of the function:
e^(2*x)/x
Identical expressions
e^(two *x)/x
e to the power of (2 multiply by x) divide by x
e to the power of (two multiply by x) divide by x
e(2*x)/x
e2*x/x
e^(2x)/x
e(2x)/x
e2x/x
e^2x/x
e^(2*x) divide by x

The derivative of the function/ e^(2*x)/x

Derivative of e^(2*x)/x

The solution

You have entered [src]

 2*x
E   
----
 x

\frac{e^{2 x}}{x}

E^(2*x)/x

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 2 x$ .
2. The derivative of $e^{u}$ is itself.
3. 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:
  
  $2 e^{2 x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
Now plug in to the quotient rule:

$\frac{2 x e^{2 x} - e^{2 x}}{x^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

\frac{2 e^{2 x}}{x} - \frac{e^{2 x}}{x^{2}}

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Derivative of e^(2*x)/x