Derivative of sqrt(x)*exp(-x)

You have entered [src]

  ___  -x
\/ x *e

$$\sqrt{x} e^{- x}$$

sqrt(x)*exp(-x)

Detail solution

Apply the quotient rule, which is:

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

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

/    ___      3        3        3   \  -x
|- \/ x  + ------- + ------ + ------|*e  
|              ___      3/2      5/2|    
\          2*\/ x    4*x      8*x   /

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

Simplify

4-я производная [src]

/  ___     2        15       3        3   \  -x
|\/ x  - ----- - ------- - ------ - ------|*e  
|          ___       7/2      5/2      3/2|    
\        \/ x    16*x      2*x      2*x   /

$$\left(\sqrt{x} - \frac{2}{\sqrt{x}} - \frac{3}{2 x^{\frac{3}{2}}} - \frac{3}{2 x^{\frac{5}{2}}} - \frac{15}{16 x^{\frac{7}{2}}}\right) e^{- x}$$

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of sqrt(x)*exp(-x)

The graph:

Piecewise:

The solution