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tgx/x
The derivative of the function/ tgx/x

Derivative of tgx/x

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
tan(x)
------
  x
tan⁡(x)x\frac{\tan{\left(x \right)}}{x} 
d /tan(x)\
--|------|
dx\  x   /
ddxtan⁡(x)x\frac{d}{d x} \frac{\tan{\left(x \right)}}{x}
Transform
Detail solution

  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=tan⁡(x)f{\left(x \right)} = \tan{\left(x \right)} and g(x)=xg{\left(x \right)} = x.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Rewrite the function to be differentiated:

      tan⁡(x)=sin⁡(x)cos⁡(x)\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}

    2. Apply the quotient rule, which is:

      ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=sin⁡(x)f{\left(x \right)} = \sin{\left(x \right)} and g(x)=cos⁡(x)g{\left(x \right)} = \cos{\left(x \right)}.

      To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. The derivative of sine is cosine:

        ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

      To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. The derivative of cosine is negative sine:

        ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

      Now plug in to the quotient rule:

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

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Apply the power rule: xx goes to 11

    Now plug in to the quotient rule:

    x(sin⁡2(x)+cos⁡2(x))cos⁡2(x)−tan⁡(x)x2\frac{\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} - \tan{\left(x \right)}}{x^{2}}

  2. Now simplify:

    x−sin⁡(2x)2x2cos⁡2(x)\frac{x - \frac{\sin{\left(2 x \right)}}{2}}{x^{2} \cos^{2}{\left(x \right)}}

The answer is:

x−sin⁡(2x)2x2cos⁡2(x)\frac{x - \frac{\sin{\left(2 x \right)}}{2}}{x^{2} \cos^{2}{\left(x \right)}}

The graph
02468-8-6-4-2-1010-500500
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
       2            
1 + tan (x)   tan(x)
----------- - ------
     x           2  
                x
tan⁡2(x)+1x−tan⁡(x)x2\frac{\tan^{2}{\left(x \right)} + 1}{x} - \frac{\tan{\left(x \right)}}{x^{2}}
Simplify
The second derivative [src] 
  /                                       2   \
  |tan(x)   /       2   \          1 + tan (x)|
2*|------ + \1 + tan (x)/*tan(x) - -----------|
  |   2                                 x     |
  \  x                                        /
-----------------------------------------------
                       x
2((tan⁡2(x)+1)tan⁡(x)−tan⁡2(x)+1x+tan⁡(x)x2)x\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{\tan^{2}{\left(x \right)} + 1}{x} + \frac{\tan{\left(x \right)}}{x^{2}}\right)}{x}
Simplify
The third derivative [src] 
  /                                             /       2   \     /       2   \       \
  |/       2   \ /         2   \   3*tan(x)   3*\1 + tan (x)/   3*\1 + tan (x)/*tan(x)|
2*|\1 + tan (x)/*\1 + 3*tan (x)/ - -------- + --------------- - ----------------------|
  |                                    3              2                   x           |
  \                                   x              x                                /
---------------------------------------------------------------------------------------
                                           x
2((tan⁡2(x)+1)(3tan⁡2(x)+1)−3(tan⁡2(x)+1)tan⁡(x)x+3(tan⁡2(x)+1)x2−3tan⁡(x)x3)x\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{x} + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)}{x^{2}} - \frac{3 \tan{\left(x \right)}}{x^{3}}\right)}{x}
Simplify
The graph
Derivative of tgx/x
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