Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of x*sqrt(3-x)
Derivative of x^6/6
Derivative of x^3-3*x
Derivative of x^2+4x
Integral of d{x}:
2*x/(x^2+1)
Graphing y =:
2*x/(x^2+1)
How do you in partial fractions?:
2*x/(x^2+1)
Identical expressions
two *x/(x^ two + one)
2 multiply by x divide by (x squared plus 1)
two multiply by x divide by (x to the power of two plus one)
2*x/(x2+1)
2*x/x2+1
2*x/(x²+1)
2*x/(x to the power of 2+1)
2x/(x^2+1)
2x/(x2+1)
2x/x2+1
2x/x^2+1
2*x divide by (x^2+1)
Similar expressions
2*x/(x^2-1)

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

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

The solution

You have entered [src]

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

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

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

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 $x^{2} + 1$ 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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

    /         2 \
    |      4*x  |
4*x*|-3 + ------|
    |          2|
    \     1 + x /
-----------------
            2    
    /     2\     
    \1 + x /

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

Simplify

The third derivative [src]

   /                   /         2 \\
   |                 2 |      2*x  ||
   |              4*x *|-1 + ------||
   |         2         |          2||
   |      4*x          \     1 + x /|
12*|-1 + ------ - ------------------|
   |          2              2      |
   \     1 + x          1 + x       /
-------------------------------------
                      2              
              /     2\               
              \1 + x /

$$\frac{12 \left(- \frac{4 x^{2} \left(\frac{2 x^{2}}{x^{2} + 1} - 1\right)}{x^{2} + 1} + \frac{4 x^{2}}{x^{2} + 1} - 1\right)}{\left(x^{2} + 1\right)^{2}}$$

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

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

The graph:

Piecewise:

The solution