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Identical expressions
[e^(-2x)]sqrt(4x- one)
[e to the power of ( minus 2x)] square root of (4x minus 1)
[e to the power of ( minus 2x)] square root of (4x minus one)
[e^(-2x)]√(4x-1)
[e(-2x)]sqrt(4x-1)
[e-2x]sqrt4x-1
[e^-2x]sqrt4x-1
Similar expressions
[e^(-2x)]sqrt(4x+1)
[e^(2x)]sqrt(4x-1)

The derivative of the function/ [e^(-2x)]sqrt(4x-1)

Derivative of [e^(-2x)]sqrt(4x-1)

The solution

You have entered [src]

 -2*x   _________
E    *\/ 4*x - 1

$$e^{- 2 x} \sqrt{4 x - 1}$$

E^(-2*x)*sqrt(4*x - 1)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sqrt{4 x - 1}$ and $g{\left(x \right)} = e^{2 x}$ .

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

$$- 2 \sqrt{4 x - 1} e^{- 2 x} + \frac{2 e^{- 2 x}}{\sqrt{4 x - 1}}$$

Simplify

The second derivative [src]

  /  __________         1              2      \  -2*x
4*|\/ -1 + 4*x  - ------------- - ------------|*e    
  |                         3/2     __________|      
  \               (-1 + 4*x)      \/ -1 + 4*x /

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

Simplify

The third derivative [src]

  /    __________         3               3              3      \  -2*x
8*|- \/ -1 + 4*x  + ------------- + ------------- + ------------|*e    
  |                           5/2             3/2     __________|      
  \                 (-1 + 4*x)      (-1 + 4*x)      \/ -1 + 4*x /

$$8 \left(- \sqrt{4 x - 1} + \frac{3}{\sqrt{4 x - 1}} + \frac{3}{\left(4 x - 1\right)^{\frac{3}{2}}} + \frac{3}{\left(4 x - 1\right)^{\frac{5}{2}}}\right) e^{- 2 x}$$

Simplify

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Derivative of [e^(-2x)]sqrt(4x-1)

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The solution