Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of cos(5-3*x)
Derivative of cos^3(t)
Derivative of asin(y^2)
Derivative of (5x^2+2)×(x^2-4)^3
Integral of d{x}:
1/(1+x^2)^2
Identical expressions
one /(one +x^ two)^ two
1 divide by (1 plus x squared ) squared
one divide by (one plus x to the power of two) to the power of two
1/(1+x2)2
1/1+x22
1/(1+x²)²
1/(1+x to the power of 2) to the power of 2
1/1+x^2^2
1 divide by (1+x^2)^2
Similar expressions
1/(1-x^2)^2

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

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

The solution

You have entered [src]

    1    
---------
        2
/     2\ 
\1 + x /

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

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

Detail solution

Let $u = \left(x^{2} + 1\right)^{2}$ .
Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} + 1\right)^{2}$ :
1. Let $u = x^{2} + 1$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} + 1\right)$ :
  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: $2 x$
  The result of the chain rule is:
  
  $2 x \left(2 x^{2} + 2\right)$
The result of the chain rule is:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  /         2 \
  |      6*x  |
4*|-1 + ------|
  |          2|
  \     1 + x /
---------------
           3   
   /     2\    
   \1 + x /

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

Simplify

The third derivative [src]

     /        2 \
     |     8*x  |
24*x*|3 - ------|
     |         2|
     \    1 + x /
-----------------
            4    
    /     2\     
    \1 + x /

$$\frac{24 x \left(- \frac{8 x^{2}}{x^{2} + 1} + 3\right)}{\left(x^{2} + 1\right)^{4}}$$

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

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

The graph:

Piecewise:

The solution