Derivative of tg2^sqrt(x)

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        \/ x 
(tan(2))

$$\tan^{\sqrt{x}}{\left(2 \right)}$$

tan(2)^(sqrt(x))

Detail solution

Let $u = \sqrt{x}$ .
$\frac{d}{d u} \tan^{u}{\left(2 \right)} = \left(\log{\left(- \tan{\left(2 \right)} \right)} + i \pi\right) \tan^{u}{\left(2 \right)}$
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:

$\frac{\left(\log{\left(- \tan{\left(2 \right)} \right)} + i \pi\right) \tan^{\sqrt{x}}{\left(2 \right)}}{2 \sqrt{x}}$

The answer is:

$\frac{\left(\log{\left(- \tan{\left(2 \right)} \right)} + i \pi\right) \tan^{\sqrt{x}}{\left(2 \right)}}{2 \sqrt{x}}$

The graph

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Plot the graph f'(x)

The first derivative [src]

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        \/ x                       
(tan(2))     *(pi*I + log(-tan(2)))
-----------------------------------
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              2*\/ x

$$\frac{\left(\log{\left(- \tan{\left(2 \right)} \right)} + i \pi\right) \tan^{\sqrt{x}}{\left(2 \right)}}{2 \sqrt{x}}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Identical expressions

Derivative of tg2^sqrt(x)

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The solution