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Derivative of:
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Identical expressions
two ^sqrt(x- one)
2 to the power of square root of (x minus 1)
two to the power of square root of (x minus one)
2^√(x-1)
2sqrt(x-1)
2sqrtx-1
2^sqrtx-1
Similar expressions
2^sqrt(x+1)

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

Derivative of 2^sqrt(x-1)

The solution

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   _______
 \/ x - 1 
2

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

2^(sqrt(x - 1))

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   _______       
 \/ x - 1        
2         *log(2)
-----------------
       _______   
   2*\/ x - 1

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

   ________ /                   2                 \       
 \/ -1 + x  |     3          log (2)      3*log(2)|       
2          *|----------- + ----------- - ---------|*log(2)
            |        5/2           3/2           2|       
            \(-1 + x)      (-1 + x)      (-1 + x) /       
----------------------------------------------------------
                            8

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

Simplify

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Derivative of 2^sqrt(x-1)

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The solution