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Identical expressions
3sqrtx*e^(-x)
3 square root of x multiply by e to the power of ( minus x)
3√x*e^(-x)
3sqrtx*e(-x)
3sqrtx*e-x
3sqrtxe^(-x)
3sqrtxe(-x)
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3sqrtxe^-x
Similar expressions
3sqrtx*e^(x)

The derivative of the function/ 3sqrtx*e^(-x)

Derivative of 3sqrtx*e^(-x)

The solution

You have entered [src]

    ___  -x
3*\/ x *e

$$3 \sqrt{x} e^{- x}$$

d /    ___  -x\
--\3*\/ x *e  /
dx

$$\frac{d}{d x} 3 \sqrt{x} e^{- x}$$

Transform

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. 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:
  
  $\left(- \sqrt{x} e^{x} + \frac{e^{x}}{2 \sqrt{x}}\right) e^{- 2 x}$
So, the result is: $3 \left(- \sqrt{x} e^{x} + \frac{e^{x}}{2 \sqrt{x}}\right) e^{- 2 x}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of 3sqrtx*e^(-x)

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Piecewise:

The solution