Derivative of 3^sqrt(x)

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   ___
 \/ x 
3

$$3^{\sqrt{x}}$$

3^(sqrt(x))

Detail solution

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

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

The answer is:

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

The graph

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The first derivative [src]

   ___       
 \/ x        
3     *log(3)
-------------
       ___   
   2*\/ x

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Identical expressions

Derivative of 3^sqrt(x)

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The solution