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Derivative of e^x*x
Differential equation:
y'
Identical expressions
y'=(x)/(sqrt4-x^ two)
y stroke first (1st) order equally (x) divide by ( square root of 4 minus x squared )
y stroke first (1st) order equally (x) divide by ( square root of 4 minus x to the power of two)
y'=(x)/(√4-x^2)
y'=(x)/(sqrt4-x2)
y'=x/sqrt4-x2
y'=(x)/(sqrt4-x²)
y'=(x)/(sqrt4-x to the power of 2)
y'=x/sqrt4-x^2
y'=(x) divide by (sqrt4-x^2)
Similar expressions
y'=(x)/(sqrt4+x^2)

The derivative of the function/ y'=(x)/(sqrt4-x^2)

Derivative of y'=(x)/(sqrt4-x^2)

The solution

You have entered [src]

    x     
----------
  ___    2
\/ 4  - x

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

d /    x     \
--|----------|
dx|  ___    2|
  \\/ 4  - x /

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

Transform

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $2 - x^{2}$ term by term:
  1. The derivative of the constant $2$ 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: $x^{2}$ goes to $2 x$
    So, the result is: $- 2 x$
  The result is: $- 2 x$
Now plug in to the quotient rule:

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

$\frac{x^{2} + 2}{\left(x^{2} - 2\right)^{2}}$

The answer is:

$\frac{x^{2} + 2}{\left(x^{2} - 2\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                     2    
    1             2*x     
---------- + -------------
  ___    2               2
\/ 4  - x    /  ___    2\ 
             \\/ 4  - x /

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                   /          2 \\
  |                 2 |       2*x  ||
  |              4*x *|-1 + -------||
  |         2         |           2||
  |      4*x          \     -2 + x /|
6*|1 - ------- + -------------------|
  |          2               2      |
  \    -2 + x          -2 + x       /
-------------------------------------
                       2             
              /      2\              
              \-2 + x /

$$\frac{6 \cdot \left(\frac{4 x^{2} \cdot \left(\frac{2 x^{2}}{x^{2} - 2} - 1\right)}{x^{2} - 2} - \frac{4 x^{2}}{x^{2} - 2} + 1\right)}{\left(x^{2} - 2\right)^{2}}$$

Simplify

3-я производная [src]

  /                   /          2 \\
  |                 2 |       2*x  ||
  |              4*x *|-1 + -------||
  |         2         |           2||
  |      4*x          \     -2 + x /|
6*|1 - ------- + -------------------|
  |          2               2      |
  \    -2 + x          -2 + x       /
-------------------------------------
                       2             
              /      2\              
              \-2 + x /

$$\frac{6 \cdot \left(\frac{4 x^{2} \cdot \left(\frac{2 x^{2}}{x^{2} - 2} - 1\right)}{x^{2} - 2} - \frac{4 x^{2}}{x^{2} - 2} + 1\right)}{\left(x^{2} - 2\right)^{2}}$$

Simplify

The graph

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Derivative of y'=(x)/(sqrt4-x^2)

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Piecewise:

The solution