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Derivative of:
Derivative of e^2*x
Derivative of cos(x)^2
Derivative of x^sin(x)
Derivative of sin(x^3)
Identical expressions
tg((x)^(one / two))/(ln(3x)/x)
tg((x) to the power of (1 divide by 2)) divide by (ln(3x) divide by x)
tg((x) to the power of (one divide by two)) divide by (ln(3x) divide by x)
tg((x)(1/2))/(ln(3x)/x)
tgx1/2/ln3x/x
tgx^1/2/ln3x/x
tg((x)^(1 divide by 2)) divide by (ln(3x) divide by x)

The derivative of the function/ tg((x)^(1/2))/(ln(3x)/x)

Derivative of tg((x)^(1/2))/(ln(3x)/x)

The solution

You have entered [src]

   /  ___\
tan\\/ x /
----------
/log(3*x)\
|--------|
\   x    /

$$\frac{\tan{\left(\sqrt{x} \right)}}{\frac{1}{x} \log{\left(3 x \right)}}$$

tan(sqrt(x))/((log(3*x)/x))

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x \tan{\left(\sqrt{x} \right)}$ and $g{\left(x \right)} = \log{\left(3 x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = \tan{\left(\sqrt{x} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Rewrite the function to be differentiated:
    
    $\tan{\left(\sqrt{x} \right)} = \frac{\sin{\left(\sqrt{x} \right)}}{\cos{\left(\sqrt{x} \right)}}$
  2. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \sin{\left(\sqrt{x} \right)}$ and $g{\left(x \right)} = \cos{\left(\sqrt{x} \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Let $u = \sqrt{x}$ .
    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 $\frac{d}{d x} \sqrt{x}$ :
      1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
      The result of the chain rule is:
      
      $\frac{\cos{\left(\sqrt{x} \right)}}{2 \sqrt{x}}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Let $u = \sqrt{x}$ .
    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 $\frac{d}{d x} \sqrt{x}$ :
      1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
      The result of the chain rule is:
      
      $- \frac{\sin{\left(\sqrt{x} \right)}}{2 \sqrt{x}}$
    Now plug in to the quotient rule:
    
    $\frac{\frac{\sin^{2}{\left(\sqrt{x} \right)}}{2 \sqrt{x}} + \frac{\cos^{2}{\left(\sqrt{x} \right)}}{2 \sqrt{x}}}{\cos^{2}{\left(\sqrt{x} \right)}}$
  The result is: $\frac{x \left(\frac{\sin^{2}{\left(\sqrt{x} \right)}}{2 \sqrt{x}} + \frac{\cos^{2}{\left(\sqrt{x} \right)}}{2 \sqrt{x}}\right)}{\cos^{2}{\left(\sqrt{x} \right)}} + \tan{\left(\sqrt{x} \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 3 x$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 x$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $3$
  The result of the chain rule is:
  
  $\frac{1}{x}$
Now plug in to the quotient rule:

$\frac{\left(\frac{x \left(\frac{\sin^{2}{\left(\sqrt{x} \right)}}{2 \sqrt{x}} + \frac{\cos^{2}{\left(\sqrt{x} \right)}}{2 \sqrt{x}}\right)}{\cos^{2}{\left(\sqrt{x} \right)}} + \tan{\left(\sqrt{x} \right)}\right) \log{\left(3 x \right)} - \tan{\left(\sqrt{x} \right)}}{\log{\left(3 x \right)}^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Identical expressions

Derivative of tg((x)^(1/2))/(ln(3x)/x)

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Piecewise:

The solution