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Identical expressions
y=e^(- four *x^ two)*sqrt(x- four *x^ two)
y equally e to the power of ( minus 4 multiply by x squared ) multiply by square root of (x minus 4 multiply by x squared )
y equally e to the power of ( minus four multiply by x to the power of two) multiply by square root of (x minus four multiply by x to the power of two)
y=e^(-4*x^2)*√(x-4*x^2)
y=e(-4*x2)*sqrt(x-4*x2)
y=e-4*x2*sqrtx-4*x2
y=e^(-4*x²)*sqrt(x-4*x²)
y=e to the power of (-4*x to the power of 2)*sqrt(x-4*x to the power of 2)
y=e^(-4x^2)sqrt(x-4x^2)
y=e(-4x2)sqrt(x-4x2)
y=e-4x2sqrtx-4x2
y=e^-4x^2sqrtx-4x^2
Similar expressions
y=e^(-4*x^2)*sqrt(x+4*x^2)
y=e^(4*x^2)*sqrt(x-4*x^2)

The derivative of the function/ y=e^(-4*x^2)*sqrt(x-4*x^2)

Derivative of y=e^(-4x^2)sqrt(x-4*x^2)

The solution

You have entered [src]

     2    __________
 -4*x    /        2 
e     *\/  x - 4*x

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

  /     2    __________\
d | -4*x    /        2 |
--\e     *\/  x - 4*x  /
dx

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

Transform

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = - 4 x^{2} + x$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(- 4 x^{2} + x\right)$ :
  1. Differentiate $- 4 x^{2} + x$ term by term:
    1. Apply the power rule: $x$ goes to $1$
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x^{2}$ goes to $2 x$
      So, the result is: $- 8 x$
    The result is: $1 - 8 x$
  The result of the chain rule is:
  
  $\frac{1 - 8 x}{2 \sqrt{- 4 x^{2} + x}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 4 x^{2}$ .
2. The derivative of $e^{u}$ is itself.
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 4 x^{2}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    So, the result is: $8 x$
  The result of the chain rule is:
  
  $8 x e^{4 x^{2}}$
Now plug in to the quotient rule:

$\left(- 8 x \sqrt{- 4 x^{2} + x} e^{4 x^{2}} + \frac{\left(1 - 8 x\right) e^{4 x^{2}}}{2 \sqrt{- 4 x^{2} + x}}\right) e^{- 8 x^{2}}$
Now simplify:

$\frac{\left(- 16 x^{2} \cdot \left(1 - 4 x\right) - 8 x + 1\right) e^{- 4 x^{2}}}{2 \sqrt{x \left(1 - 4 x\right)}}$

The answer is:

$\frac{\left(- 16 x^{2} \cdot \left(1 - 4 x\right) - 8 x + 1\right) e^{- 4 x^{2}}}{2 \sqrt{x \left(1 - 4 x\right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                 2                           
             -4*x           __________      2
(1/2 - 4*x)*e              /        2   -4*x 
------------------ - 8*x*\/  x - 4*x  *e     
     __________                              
    /        2                               
  \/  x - 4*x

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

Simplify

The second derivative [src]

/                                                2                    \       
|                                      (-1 + 8*x)                     |       
|                                 16 - ------------                   |      2
|    _____________ /        2\         x*(-1 + 4*x)     8*x*(-1 + 8*x)|  -4*x 
|8*\/ x*(1 - 4*x) *\-1 + 8*x / - ------------------- + ---------------|*e     
|                                    _______________     _____________|       
\                                4*\/ -x*(-1 + 4*x)    \/ x*(1 - 4*x) /

$$\left(\frac{8 x \left(8 x - 1\right)}{\sqrt{x \left(1 - 4 x\right)}} + 8 \sqrt{x \left(1 - 4 x\right)} \left(8 x^{2} - 1\right) - \frac{16 - \frac{\left(8 x - 1\right)^{2}}{x \left(4 x - 1\right)}}{4 \sqrt{- x \left(4 x - 1\right)}}\right) e^{- 4 x^{2}}$$

Simplify

The third derivative [src]

/                                                                     /               2 \                /               2 \\       
|                                                                     |     (-1 + 8*x)  |                |     (-1 + 8*x)  ||       
|                                                   /        2\   6*x*|16 - ------------|   3*(-1 + 8*x)*|16 - ------------||      2
|         _____________ /        2\   12*(-1 + 8*x)*\-1 + 8*x /       \     x*(-1 + 4*x)/                \     x*(-1 + 4*x)/|  -4*x 
|- 64*x*\/ x*(1 - 4*x) *\-3 + 8*x / - ------------------------- + ----------------------- - --------------------------------|*e     
|                                            _____________             _______________                             3/2      |       
\                                          \/ x*(1 - 4*x)            \/ -x*(-1 + 4*x)             8*(-x*(-1 + 4*x))         /

$$\left(- 64 x \sqrt{x \left(1 - 4 x\right)} \left(8 x^{2} - 3\right) + \frac{6 x \left(16 - \frac{\left(8 x - 1\right)^{2}}{x \left(4 x - 1\right)}\right)}{\sqrt{- x \left(4 x - 1\right)}} - \frac{3 \cdot \left(16 - \frac{\left(8 x - 1\right)^{2}}{x \left(4 x - 1\right)}\right) \left(8 x - 1\right)}{8 \left(- x \left(4 x - 1\right)\right)^{\frac{3}{2}}} - \frac{12 \cdot \left(8 x - 1\right) \left(8 x^{2} - 1\right)}{\sqrt{x \left(1 - 4 x\right)}}\right) e^{- 4 x^{2}}$$

Simplify

The graph

Derivative of y=e^(-4*x^2)*sqrt(x-4*x^2)

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Identical expressions

Similar expressions

Derivative of y=e^(-4x^2)sqrt(x-4*x^2)

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of y=e^(-4*x^2)*sqrt(x-4*x^2)

The graph:

Piecewise:

The solution

Derivative of y=e^(-4x^2)sqrt(x-4*x^2)