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Identical expressions
(√x^ three + four)*e^(-2x)
(√x cubed plus 4) multiply by e to the power of ( minus 2x)
(√x to the power of three plus four) multiply by e to the power of ( minus 2x)
(√x3+4)*e(-2x)
√x3+4*e-2x
(√x³+4)*e^(-2x)
(√x to the power of 3+4)*e to the power of (-2x)
(√x^3+4)e^(-2x)
(√x3+4)e(-2x)
√x3+4e-2x
√x^3+4e^-2x
Similar expressions
(√x^3-4)*e^(-2x)
(√x^3+4)*e^(2x)

The derivative of the function/ (√x^3+4)*e^(-2x)

Derivative of (√x^3+4)*e^(-2x)

The solution

You have entered [src]

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

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

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

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

Transform

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x^{\frac{3}{2}} + 4$ and $g{\left(x \right)} = e^{2 x}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x^{\frac{3}{2}} + 4$ term by term:
  1. The derivative of the constant $4$ is zero.
  2. Apply the power rule: $x^{\frac{3}{2}}$ goes to $\frac{3 \sqrt{x}}{2}$
  The result is: $\frac{3 \sqrt{x}}{2}$
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:

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

$\frac{\left(- 4 x^{\frac{3}{2}} + 3 \sqrt{x} - 16\right) e^{- 2 x}}{2}$

The answer is:

$\frac{\left(- 4 x^{\frac{3}{2}} + 3 \sqrt{x} - 16\right) e^{- 2 x}}{2}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

/         ___      3/2      3   \  -2*x
|16 - 6*\/ x  + 4*x    + -------|*e    
|                            ___|      
\                        4*\/ x /

$$\left(4 x^{\frac{3}{2}} - 6 \sqrt{x} + 16 + \frac{3}{4 \sqrt{x}}\right) e^{- 2 x}$$

Simplify

The third derivative [src]

/         3/2        ___      9        3   \  -2*x
|-32 - 8*x    + 18*\/ x  - ------- - ------|*e    
|                              ___      3/2|      
\                          2*\/ x    8*x   /

$$\left(- 8 x^{\frac{3}{2}} + 18 \sqrt{x} - 32 - \frac{9}{2 \sqrt{x}} - \frac{3}{8 x^{\frac{3}{2}}}\right) e^{- 2 x}$$

Simplify

The graph

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Derivative of (√x^3+4)*e^(-2x)

The graph:

Piecewise:

The solution