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Integral of d{x}:
x*sqrt(1+x^2)
Limit of the function:
x*sqrt(1+x^2)
Identical expressions
x*sqrt(one +x^ two)
x multiply by square root of (1 plus x squared )
x multiply by square root of (one plus x to the power of two)
x*√(1+x^2)
x*sqrt(1+x2)
x*sqrt1+x2
x*sqrt(1+x²)
x*sqrt(1+x to the power of 2)
xsqrt(1+x^2)
xsqrt(1+x2)
xsqrt1+x2
xsqrt1+x^2
Similar expressions
x*sqrt(1-x^2)

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

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

The solution

You have entered [src]

     ________
    /      2 
x*\/  1 + x

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

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

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

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
$g{\left(x \right)} = \sqrt{x^{2} + 1}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = x^{2} + 1$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
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:
  
  $\frac{x}{\sqrt{x^{2} + 1}}$
The result is: $\frac{x^{2}}{\sqrt{x^{2} + 1}} + \sqrt{x^{2} + 1}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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

The graph:

Piecewise:

The solution