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exp^(one -2x)/√x
exponent of to the power of (1 minus 2x) divide by √x
exponent of to the power of (one minus 2x) divide by √x
exp(1-2x)/√x
exp1-2x/√x
exp^1-2x/√x
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Similar expressions
exp^(1+2x)/√x

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

Derivative of exp^(1-2x)/√x

The solution

You have entered [src]

 1 - 2*x
E       
--------
   ___  
 \/ x

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

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

Detail solution

Apply the quotient rule, which is:

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

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

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

$\frac{- 2 \sqrt{x} e^{1 - 2 x} - \frac{e^{1 - 2 x}}{2 \sqrt{x}}}{x}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     1 - 2*x    1 - 2*x
  2*e          e       
- ---------- - --------
      ___          3/2 
    \/ x        2*x

$$- \frac{2 e^{1 - 2 x}}{\sqrt{x}} - \frac{e^{1 - 2 x}}{2 x^{\frac{3}{2}}}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

 /    6    9      15 \  1 - 2*x 
-|8 + - + ---- + ----|*e        
 |    x      2      3|          
 \        2*x    8*x /          
--------------------------------
               ___              
             \/ x

$$- \frac{\left(8 + \frac{6}{x} + \frac{9}{2 x^{2}} + \frac{15}{8 x^{3}}\right) e^{1 - 2 x}}{\sqrt{x}}$$

Simplify

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Derivative of exp^(1-2x)/√x

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The solution