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Integral of d{x}:
2^x/(x+1)
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Similar expressions
2^x/(x-1)

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

Derivative of 2^x/(x+1)

The solution

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   x 
  2  
-----
x + 1

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

2^x/(x + 1)

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 $g{\left(x \right)} = x + 1$ .

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 $x + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Apply the power rule: $x$ goes to $1$
  The result is: $1$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of 2^x/(x+1)

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The solution