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Derivative of:
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Derivative of x^x^2
Derivative of -e^-x
Derivative of (x)^(1/2)
Identical expressions
(one +x)*sqrt(one +sqrtx)
(1 plus x) multiply by square root of (1 plus square root of x)
(one plus x) multiply by square root of (one plus square root of x)
(1+x)*√(1+√x)
(1+x)sqrt(1+sqrtx)
1+xsqrt1+sqrtx
Similar expressions
(1-x)*sqrt(1+sqrtx)
(1+x)*sqrt(1-sqrtx)

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

Derivative of (1+x)*sqrt(1+sqrtx)

The solution

You have entered [src]

           ___________
          /       ___ 
(1 + x)*\/  1 + \/ x

$$\sqrt{\sqrt{x} + 1} \left(x + 1\right)$$

(1 + x)*sqrt(1 + sqrt(x))

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x + 1$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
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: $1$
$g{\left(x \right)} = \sqrt{\sqrt{x} + 1}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \sqrt{x} + 1$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\sqrt{x} + 1\right)$ :
  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 result is: $\sqrt{\sqrt{x} + 1} + \frac{x + 1}{4 \sqrt{x} \sqrt{\sqrt{x} + 1}}$
Now simplify:

$\frac{4 \sqrt{x} + 5 x + 1}{4 \sqrt{x} \sqrt{\sqrt{x} + 1}}$

The answer is:

$\frac{4 \sqrt{x} + 5 x + 1}{4 \sqrt{x} \sqrt{\sqrt{x} + 1}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  8             / 2           1      \
----- - (1 + x)*|---- + -------------|
  ___           | 3/2     /      ___\|
\/ x            \x      x*\1 + \/ x //
--------------------------------------
                ___________           
               /       ___            
          16*\/  1 + \/ x

$$\frac{- \left(x + 1\right) \left(\frac{1}{x \left(\sqrt{x} + 1\right)} + \frac{2}{x^{\frac{3}{2}}}\right) + \frac{8}{\sqrt{x}}}{16 \sqrt{\sqrt{x} + 1}}$$

Simplify

The third derivative [src]

  /   8             / 4             1                 2       \         4      \
3*|- ---- + (1 + x)*|---- + ----------------- + --------------| - -------------|
  |   3/2           | 5/2                   2    2 /      ___\|     /      ___\|
  |  x              |x       3/2 /      ___\    x *\1 + \/ x /|   x*\1 + \/ x /|
  \                 \       x   *\1 + \/ x /                  /                /
--------------------------------------------------------------------------------
                                     ___________                                
                                    /       ___                                 
                               64*\/  1 + \/ x

$$\frac{3 \left(\left(x + 1\right) \left(\frac{2}{x^{2} \left(\sqrt{x} + 1\right)} + \frac{1}{x^{\frac{3}{2}} \left(\sqrt{x} + 1\right)^{2}} + \frac{4}{x^{\frac{5}{2}}}\right) - \frac{4}{x \left(\sqrt{x} + 1\right)} - \frac{8}{x^{\frac{3}{2}}}\right)}{64 \sqrt{\sqrt{x} + 1}}$$

Simplify

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Derivative of (1+x)*sqrt(1+sqrtx)

The graph:

Piecewise:

The solution