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

You entered:

2*x/(1+x)

What you mean?

2*x/(1 + x)

2*x/1 + x

2*x/(1 + x)

2*x/1 + x

2*x/1 + x

Derivative of 2*x/(1+x)

The solution

You have entered [src]

 2*x 
-----
1 + x

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

d / 2*x \
--|-----|
dx\1 + x/

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

Transform

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. 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 $g{\left(x \right)} = x + 1$ .
  
  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 $x + 1$ 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:
  
  $\frac{1}{\left(x + 1\right)^{2}}$
So, the result is: $\frac{2}{\left(x + 1\right)^{2}}$

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

   /      x  \
12*|1 - -----|
   \    1 + x/
--------------
          3   
   (1 + x)

$$\frac{12 \left(- \frac{x}{x + 1} + 1\right)}{\left(x + 1\right)^{3}}$$

Simplify

The graph

Derivative of 2*x/(1+x)