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

Derivative of 2*x/(x+1)

The solution

You have entered [src]

 2*x 
-----
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. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $x$ goes to $1$
  So, the result is: $2$
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:

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

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

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

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/(x+1)