Derivative of tg(x^(sqrt(x)))

The solution

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tan\x     /

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

Transform

Detail solution

Rewrite the function to be differentiated:

$\tan{\left(x^{\sqrt{x}} \right)} = \frac{\sin{\left(x^{\sqrt{x}} \right)}}{\cos{\left(x^{\sqrt{x}} \right)}}$
Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \sin{\left(x^{\sqrt{x}} \right)}$ and $g{\left(x \right)} = \cos{\left(x^{\sqrt{x}} \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = x^{\sqrt{x}}$ .
2. The derivative of sine is cosine:
  
  $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} x^{\sqrt{x}}$ :
  1. Don't know the steps in finding this derivative.
    
    But the derivative is
    
    $x^{\frac{\sqrt{x}}{2}} \left(\log{\left(\sqrt{x} \right)} + 1\right)$
  The result of the chain rule is:
  
  $x^{\frac{\sqrt{x}}{2}} \left(\log{\left(\sqrt{x} \right)} + 1\right) \cos{\left(x^{\sqrt{x}} \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = x^{\sqrt{x}}$ .
2. The derivative of cosine is negative sine:
  
  $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} x^{\sqrt{x}}$ :
  1. Don't know the steps in finding this derivative.
    
    But the derivative is
    
    $x^{\frac{\sqrt{x}}{2}} \left(\log{\left(\sqrt{x} \right)} + 1\right)$
  The result of the chain rule is:
  
  $- x^{\frac{\sqrt{x}}{2}} \left(\log{\left(\sqrt{x} \right)} + 1\right) \sin{\left(x^{\sqrt{x}} \right)}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                          /                                                                   ___               /        /   ___\\          ___                   /   ___\        ___                  /   ___\        ___                        /   ___\\
   ___ /        /   ___\\ |                     3                                         2*\/ x              3 |       2| \/ x ||      2*\/ x              3    2| \/ x |      \/ x              3    | \/ x |      \/ x                         | \/ x ||
 \/ x  |       2| \/ x || |   2     (2 + log(x))    3*log(x)   3*(2 + log(x))*log(x)   2*x       *(2 + log(x)) *\1 + tan \x     //   4*x       *(2 + log(x)) *tan \x     /   6*x     *(2 + log(x)) *tan\x     /   6*x     *(2 + log(x))*log(x)*tan\x     /|
x     *\1 + tan \x     //*|- ---- + ------------- + -------- - --------------------- + ------------------------------------------- + ------------------------------------- + ---------------------------------- - ----------------------------------------|
                          |   5/2         3/2          5/2                2                                 3/2                                        3/2                                   3/2                                      2                   |
                          \  x           x            x                  x                                 x                                          x                                     x                                        x                    /
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                                                                                                                             8

$$\frac{x^{\sqrt{x}} \left(\tan^{2}{\left(x^{\sqrt{x}} \right)} + 1\right) \left(- \frac{6 x^{\sqrt{x}} \left(\log{\left(x \right)} + 2\right) \log{\left(x \right)} \tan{\left(x^{\sqrt{x}} \right)}}{x^{2}} - \frac{3 \left(\log{\left(x \right)} + 2\right) \log{\left(x \right)}}{x^{2}} + \frac{2 x^{2 \sqrt{x}} \left(\log{\left(x \right)} + 2\right)^{3} \left(\tan^{2}{\left(x^{\sqrt{x}} \right)} + 1\right)}{x^{\frac{3}{2}}} + \frac{4 x^{2 \sqrt{x}} \left(\log{\left(x \right)} + 2\right)^{3} \tan^{2}{\left(x^{\sqrt{x}} \right)}}{x^{\frac{3}{2}}} + \frac{6 x^{\sqrt{x}} \left(\log{\left(x \right)} + 2\right)^{3} \tan{\left(x^{\sqrt{x}} \right)}}{x^{\frac{3}{2}}} + \frac{\left(\log{\left(x \right)} + 2\right)^{3}}{x^{\frac{3}{2}}} + \frac{3 \log{\left(x \right)}}{x^{\frac{5}{2}}} - \frac{2}{x^{\frac{5}{2}}}\right)}{8}$$

Simplify

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Derivative of tg(x^(sqrt(x)))

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