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Derivative of:
Derivative of -3/x
Derivative of x^(2*x)
Derivative of 3*x
Derivative of 2/sqrt(x)
Identical expressions
ln(tg(x)/ two ^(one / two)*x^ two)
ln(tg(x) divide by 2 to the power of (1 divide by 2) multiply by x squared )
ln(tg(x) divide by two to the power of (one divide by two) multiply by x to the power of two)
ln(tg(x)/2(1/2)*x2)
lntgx/21/2*x2
ln(tg(x)/2^(1/2)*x²)
ln(tg(x)/2 to the power of (1/2)*x to the power of 2)
ln(tg(x)/2^(1/2)x^2)
ln(tg(x)/2(1/2)x2)
lntgx/21/2x2
lntgx/2^1/2x^2
ln(tg(x) divide by 2^(1 divide by 2)*x^2)

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

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

The solution

You have entered [src]

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

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

Transform

Detail solution

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

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

$\frac{2}{\sin{\left(2 x \right)}} + \frac{2}{x}$

The answer is:

$\frac{2}{\sin{\left(2 x \right)}} + \frac{2}{x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

    /             /       2   \\     /    /       2   \    2 /       2   \                \   /       2   \ /             /       2   \\
  2*\2*tan(x) + x*\1 + tan (x)//   2*\2*x*\1 + tan (x)/ + x *\1 + tan (x)/*tan(x) + tan(x)/   \1 + tan (x)/*\2*tan(x) + x*\1 + tan (x)//
- ------------------------------ + -------------------------------------------------------- - ------------------------------------------
                x                                             x                                                 tan(x)                  
----------------------------------------------------------------------------------------------------------------------------------------
                                                                x*tan(x)

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

Simplify

The third derivative [src]

  /                                                                                                                                                                                                                           2                                                                                                                                                     \
  |                                                 /    /       2   \    2 /       2   \                \     /             /       2   \\   /       2   \ /     2 /       2   \      2    2                \   /       2   \  /             /       2   \\     /       2   \ /    /       2   \    2 /       2   \                \     /       2   \ /             /       2   \\|
  |  /       2   \ /             /       2   \\   4*\2*x*\1 + tan (x)/ + x *\1 + tan (x)/*tan(x) + tan(x)/   3*\2*tan(x) + x*\1 + tan (x)//   \1 + tan (x)/*\3 + x *\1 + tan (x)/ + 2*x *tan (x) + 6*x*tan(x)/   \1 + tan (x)/ *\2*tan(x) + x*\1 + tan (x)//   2*\1 + tan (x)/*\2*x*\1 + tan (x)/ + x *\1 + tan (x)/*tan(x) + tan(x)/   2*\1 + tan (x)/*\2*tan(x) + x*\1 + tan (x)//|
2*|- \1 + tan (x)/*\2*tan(x) + x*\1 + tan (x)// - -------------------------------------------------------- + ------------------------------ + ---------------------------------------------------------------- + ------------------------------------------- - ---------------------------------------------------------------------- + --------------------------------------------|
  |                                                                           2                                             2                                                x                                                        2                                                       x*tan(x)                                                    x*tan(x)                  |
  \                                                                          x                                             x                                                                                                       tan (x)                                                                                                                                          /
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                       x*tan(x)

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

Simplify

The graph

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Identical expressions

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

The graph:

Piecewise:

The solution