Derivative of xtgx/4

The solution

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x*tan(x)
--------
   4

$$\frac{x \tan{\left(x \right)}}{4}$$

(x*tan(x))/4

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
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 $1$
  $g{\left(x \right)} = \tan{\left(x \right)}$ ; 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: $\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \tan{\left(x \right)}$
So, the result is: $\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{4 \cos^{2}{\left(x \right)}} + \frac{\tan{\left(x \right)}}{4}$
Now simplify:

$\frac{x + \frac{\sin{\left(2 x \right)}}{2}}{4 \cos^{2}{\left(x \right)}}$

The answer is:

$\frac{x + \frac{\sin{\left(2 x \right)}}{2}}{4 \cos^{2}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Identical expressions

Derivative of xtgx/4

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The solution