Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of 2*x*sin(x)
Derivative of (x^2+3*x-10)^(5/2)
Derivative of (x+1)*(x+2)^2
Derivative of -x+1
Integral of d{x}:
x^2/(x+1)^2
How do you in partial fractions?:
x^2/(x+1)^2
Identical expressions
x^ two /(x+ one)^ two
x squared divide by (x plus 1) squared
x to the power of two divide by (x plus one) to the power of two
x2/(x+1)2
x2/x+12
x²/(x+1)²
x to the power of 2/(x+1) to the power of 2
x^2/x+1^2
x^2 divide by (x+1)^2
Similar expressions
x^2/(x-1)^2

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

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

The solution

You have entered [src]

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

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

x^2/(x + 1)^2

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

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

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

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

$\frac{2 x}{x^{3} + 3 x^{2} + 3 x + 1}$

The answer is:

$\frac{2 x}{x^{3} + 3 x^{2} + 3 x + 1}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

            2           
  2*x      x *(-2 - 2*x)
-------- + -------------
       2             4  
(x + 1)       (x + 1)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

   /          2          \
   |       2*x       3*x |
12*|-1 - -------- + -----|
   |            2   1 + x|
   \     (1 + x)         /
--------------------------
                3         
         (1 + x)

$$\frac{12 \left(- \frac{2 x^{2}}{\left(x + 1\right)^{2}} + \frac{3 x}{x + 1} - 1\right)}{\left(x + 1\right)^{3}}$$

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

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

The graph:

Piecewise:

The solution