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The derivative of the function/ (tan(3x))/(tan(x))

Derivative of (tan(3x))/(tan(x))

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

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tan(3*x)
--------
 tan(x)
tan⁡(3x)tan⁡(x)\frac{\tan{\left(3 x \right)}}{\tan{\left(x \right)}} 
tan(3*x)/tan(x)
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⁡(3x)f{\left(x \right)} = \tan{\left(3 x \right)} and g(x)=tan⁡(x)g{\left(x \right)} = \tan{\left(x \right)}.

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

    1. Rewrite the function to be differentiated:

      tan⁡(3x)=sin⁡(3x)cos⁡(3x)\tan{\left(3 x \right)} = \frac{\sin{\left(3 x \right)}}{\cos{\left(3 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⁡(3x)f{\left(x \right)} = \sin{\left(3 x \right)} and g(x)=cos⁡(3x)g{\left(x \right)} = \cos{\left(3 x \right)}.

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

      1. Let u=3xu = 3 x.

      2. The derivative of sine is cosine:

        ddusin⁡(u)=cos⁡(u)\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}

      3. Then, apply the chain rule. Multiply by ddx3x\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: xx goes to 11

          So, the result is: 33

        The result of the chain rule is:

        3cos⁡(3x)3 \cos{\left(3 x \right)}

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

      1. Let u=3xu = 3 x.

      2. The derivative of cosine is negative sine:

        dducos⁡(u)=−sin⁡(u)\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}

      3. Then, apply the chain rule. Multiply by ddx3x\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: xx goes to 11

          So, the result is: 33

        The result of the chain rule is:

        −3sin⁡(3x)- 3 \sin{\left(3 x \right)}

      Now plug in to the quotient rule:

      3sin⁡2(3x)+3cos⁡2(3x)cos⁡2(3x)\frac{3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}}

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

    1. ddxtan⁡(x)=1cos⁡2(x)\frac{d}{d x} \tan{\left(x \right)} = \frac{1}{\cos^{2}{\left(x \right)}}

    Now plug in to the quotient rule:

    −(sin⁡2(x)+cos⁡2(x))tan⁡(3x)cos⁡2(x)+(3sin⁡2(3x)+3cos⁡2(3x))tan⁡(x)cos⁡2(3x)tan⁡2(x)\frac{- \frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(3 x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\left(3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}\right) \tan{\left(x \right)}}{\cos^{2}{\left(3 x \right)}}}{\tan^{2}{\left(x \right)}}

  2. Now simplify:

    3cos⁡2(3x)tan⁡(x)−tan⁡(3x)sin⁡2(x)\frac{3}{\cos^{2}{\left(3 x \right)} \tan{\left(x \right)}} - \frac{\tan{\left(3 x \right)}}{\sin^{2}{\left(x \right)}}

The answer is:

3cos⁡2(3x)tan⁡(x)−tan⁡(3x)sin⁡2(x)\frac{3}{\cos^{2}{\left(3 x \right)} \tan{\left(x \right)}} - \frac{\tan{\left(3 x \right)}}{\sin^{2}{\left(x \right)}}

The graph
02468-8-6-4-2-1010-500000500000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
         2        /        2   \         
3 + 3*tan (3*x)   \-1 - tan (x)/*tan(3*x)
--------------- + -----------------------
     tan(x)                  2           
                          tan (x)
(−tan⁡2(x)−1)tan⁡(3x)tan⁡2(x)+3tan⁡2(3x)+3tan⁡(x)\frac{\left(- \tan^{2}{\left(x \right)} - 1\right) \tan{\left(3 x \right)}}{\tan^{2}{\left(x \right)}} + \frac{3 \tan^{2}{\left(3 x \right)} + 3}{\tan{\left(x \right)}}
Simplify
The second derivative [src] 
  /                                           /            2   \              /       2   \ /       2     \\
  |  /       2     \            /       2   \ |     1 + tan (x)|            3*\1 + tan (x)/*\1 + tan (3*x)/|
2*|9*\1 + tan (3*x)/*tan(3*x) + \1 + tan (x)/*|-1 + -----------|*tan(3*x) - -------------------------------|
  |                                           |          2     |                         tan(x)            |
  \                                           \       tan (x)  /                                           /
------------------------------------------------------------------------------------------------------------
                                                   tan(x)
2((tan⁡2(x)+1tan⁡2(x)−1)(tan⁡2(x)+1)tan⁡(3x)−3(tan⁡2(x)+1)(tan⁡2(3x)+1)tan⁡(x)+9(tan⁡2(3x)+1)tan⁡(3x))tan⁡(x)\frac{2 \left(\left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)}} - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(3 x \right)} - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\tan^{2}{\left(3 x \right)} + 1\right)}{\tan{\left(x \right)}} + 9 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(3 x \right)}\right)}{\tan{\left(x \right)}}
Simplify
The third derivative [src] 
  /                                                                                                                                                                                      /            2   \\
  |                                                                                                                                                        /       2   \ /       2     \ |     1 + tan (x)||
  |  /                               2                  3\                                                                                               9*\1 + tan (x)/*\1 + tan (3*x)/*|-1 + -----------||
  |  |                  /       2   \      /       2   \ |               /       2     \ /         2     \      /       2   \ /       2     \                                            |          2     ||
  |  |         2      5*\1 + tan (x)/    3*\1 + tan (x)/ |            27*\1 + tan (3*x)/*\1 + 3*tan (3*x)/   27*\1 + tan (x)/*\1 + tan (3*x)/*tan(3*x)                                   \       tan (x)  /|
2*|- |2 + 2*tan (x) - ---------------- + ----------------|*tan(3*x) + ------------------------------------ - ----------------------------------------- + --------------------------------------------------|
  |  |                       2                  4        |                           tan(x)                                      2                                             tan(x)                      |
  \  \                    tan (x)            tan (x)     /                                                                    tan (x)                                                                      /
2(9(tan⁡2(x)+1tan⁡2(x)−1)(tan⁡2(x)+1)(tan⁡2(3x)+1)tan⁡(x)−27(tan⁡2(x)+1)(tan⁡2(3x)+1)tan⁡(3x)tan⁡2(x)+27(tan⁡2(3x)+1)(3tan⁡2(3x)+1)tan⁡(x)−(3(tan⁡2(x)+1)3tan⁡4(x)−5(tan⁡2(x)+1)2tan⁡2(x)+2tan⁡2(x)+2)tan⁡(3x))2 \left(\frac{9 \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)}} - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) \left(\tan^{2}{\left(3 x \right)} + 1\right)}{\tan{\left(x \right)}} - \frac{27 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(3 x \right)}}{\tan^{2}{\left(x \right)}} + \frac{27 \left(\tan^{2}{\left(3 x \right)} + 1\right) \left(3 \tan^{2}{\left(3 x \right)} + 1\right)}{\tan{\left(x \right)}} - \left(\frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)^{3}}{\tan^{4}{\left(x \right)}} - \frac{5 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{2}{\left(x \right)}} + 2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(3 x \right)}\right)
Simplify
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