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2^x/(1+x^2)
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Similar expressions
2^x/(1-x^2)

The derivative of the function/ 2^x/(1+x^2)

Derivative of 2^x/(1+x^2)

The solution

You have entered [src]

   x  
  2   
------
     2
1 + x

$$\frac{2^{x}}{x^{2} + 1}$$

2^x/(1 + x^2)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 2^{x}$ and $g{\left(x \right)} = x^{2} + 1$ .

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

$\frac{- 2 \cdot 2^{x} x + 2^{x} \left(x^{2} + 1\right) \log{\left(2 \right)}}{\left(x^{2} + 1\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 x                 x 
2 *log(2)     2*x*2  
--------- - ---------
       2            2
  1 + x     /     2\ 
            \1 + x /

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

Simplify

The second derivative [src]

   /            /         2 \             \
   |            |      4*x  |             |
   |          2*|-1 + ------|             |
   |            |          2|             |
 x |   2        \     1 + x /   4*x*log(2)|
2 *|log (2) + --------------- - ----------|
   |                    2              2  |
   \               1 + x          1 + x   /
-------------------------------------------
                        2                  
                   1 + x

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

Simplify

The third derivative [src]

   /               /         2 \                   /         2 \       \
   |               |      2*x  |                   |      4*x  |       |
   |          24*x*|-1 + ------|                 6*|-1 + ------|*log(2)|
   |               |          2|          2        |          2|       |
 x |   3           \     1 + x /   6*x*log (2)     \     1 + x /       |
2 *|log (2) - ------------------ - ----------- + ----------------------|
   |                      2                2                  2        |
   |              /     2\            1 + x              1 + x         |
   \              \1 + x /                                             /
------------------------------------------------------------------------
                                      2                                 
                                 1 + x

$$\frac{2^{x} \left(- \frac{6 x \log{\left(2 \right)}^{2}}{x^{2} + 1} - \frac{24 x \left(\frac{2 x^{2}}{x^{2} + 1} - 1\right)}{\left(x^{2} + 1\right)^{2}} + \log{\left(2 \right)}^{3} + \frac{6 \left(\frac{4 x^{2}}{x^{2} + 1} - 1\right) \log{\left(2 \right)}}{x^{2} + 1}\right)}{x^{2} + 1}$$

Simplify

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Derivative of 2^x/(1+x^2)

The graph:

Piecewise:

The solution