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Identical expressions
sqrt(one +y^ two)
square root of (1 plus y squared )
square root of (one plus y to the power of two)
√(1+y^2)
sqrt(1+y2)
sqrt1+y2
sqrt(1+y²)
sqrt(1+y to the power of 2)
sqrt1+y^2
Similar expressions
sqrt(1-y^2)

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

Derivative of sqrt(1+y^2)

The solution

You have entered [src]

   ________
  /      2 
\/  1 + y

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

sqrt(1 + y^2)

Detail solution

Let $u = y^{2} + 1$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d y} \left(y^{2} + 1\right)$ :
1. Differentiate $y^{2} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Apply the power rule: $y^{2}$ goes to $2 y$
  The result is: $2 y$
The result of the chain rule is:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     y     
-----------
   ________
  /      2 
\/  1 + y

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

Simplify

The second derivative [src]

        2  
       y   
 1 - ------
          2
     1 + y 
-----------
   ________
  /      2 
\/  1 + y

$$\frac{- \frac{y^{2}}{y^{2} + 1} + 1}{\sqrt{y^{2} + 1}}$$

Simplify

The third derivative [src]

    /        2  \
    |       y   |
3*y*|-1 + ------|
    |          2|
    \     1 + y /
-----------------
           3/2   
   /     2\      
   \1 + y /

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

Simplify

The graph

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Identical expressions

Similar expressions

Derivative of sqrt(1+y^2)

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The solution