Derivative of sqrt(1+sqrt(x))

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   ___________
  /       ___ 
\/  1 + \/ x

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

  /   ___________\
d |  /       ___ |
--\\/  1 + \/ x  /
dx

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

Transform

Detail solution

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

$\frac{1}{4 \sqrt{x} \sqrt{\sqrt{x} + 1}}$

The answer is:

$\frac{1}{4 \sqrt{x} \sqrt{\sqrt{x} + 1}}$

The graph

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The first derivative [src]

          1           
----------------------
           ___________
    ___   /       ___ 
4*\/ x *\/  1 + \/ x

$$\frac{1}{4 \sqrt{x} \sqrt{\sqrt{x} + 1}}$$

Simplify

The second derivative [src]

 / 2           1      \ 
-|---- + -------------| 
 | 3/2     /      ___\| 
 \x      x*\1 + \/ x // 
------------------------
         ___________    
        /       ___     
   16*\/  1 + \/ x

$$- \frac{\frac{1}{x \left(\sqrt{x} + 1\right)} + \frac{2}{x^{\frac{3}{2}}}}{16 \sqrt{\sqrt{x} + 1}}$$

Simplify

The third derivative [src]

  / 4             1                 2       \
3*|---- + ----------------- + --------------|
  | 5/2                   2    2 /      ___\|
  |x       3/2 /      ___\    x *\1 + \/ x /|
  \       x   *\1 + \/ x /                  /
---------------------------------------------
                    ___________              
                   /       ___               
              64*\/  1 + \/ x

$$\frac{3 \cdot \left(\frac{2}{x^{2} \left(\sqrt{x} + 1\right)} + \frac{1}{x^{\frac{3}{2}} \left(\sqrt{x} + 1\right)^{2}} + \frac{4}{x^{\frac{5}{2}}}\right)}{64 \sqrt{\sqrt{x} + 1}}$$

Simplify

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Derivative of sqrt(1+sqrt(x))

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The solution