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y=tg^(3)(x)-3tg(x)+3x
The derivative of the function/ y=tg^(3)(x)-3tg(x)+3x

Derivative of y=tg^(3)(x)-3tg(x)+3x

Function f() - derivative -N order at the point
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   3                    
tan (x) - 3*tan(x) + 3*x
3x+(tan⁡3(x)−3tan⁡(x))3 x + \left(\tan^{3}{\left(x \right)} - 3 \tan{\left(x \right)}\right) 
tan(x)^3 - 3*tan(x) + 3*x
Detail solution

  1. Differentiate 3x+(tan⁡3(x)−3tan⁡(x))3 x + \left(\tan^{3}{\left(x \right)} - 3 \tan{\left(x \right)}\right) term by term:

    1. Differentiate tan⁡3(x)−3tan⁡(x)\tan^{3}{\left(x \right)} - 3 \tan{\left(x \right)} term by term:

      1. Let u=tan⁡(x)u = \tan{\left(x \right)}.

      2. Apply the power rule: u3u^{3} goes to 3u23 u^{2}

      3. Then, apply the chain rule. Multiply by ddxtan⁡(x)\frac{d}{d x} \tan{\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)}}

        The result of the chain rule is:

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

      4. The derivative of a constant times a function is the constant times the derivative of the function.

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

        So, the result is: −3(sin⁡2(x)+cos⁡2(x))cos⁡2(x)- \frac{3 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}}

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

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

  2. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-10104000000-2000000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
       2         2    /         2   \
- 3*tan (x) + tan (x)*\3 + 3*tan (x)/
(3tan⁡2(x)+3)tan⁡2(x)−3tan⁡2(x)\left(3 \tan^{2}{\left(x \right)} + 3\right) \tan^{2}{\left(x \right)} - 3 \tan^{2}{\left(x \right)}
Simplify
The second derivative [src] 
      3    /       2   \
12*tan (x)*\1 + tan (x)/
12(tan⁡2(x)+1)tan⁡3(x)12 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{3}{\left(x \right)}
Simplify
The third derivative [src] 
                /                  2                                                  \
  /       2   \ |     /       2   \         2           4           2    /       2   \|
6*\1 + tan (x)/*\-1 + \1 + tan (x)/  - 3*tan (x) + 2*tan (x) + 7*tan (x)*\1 + tan (x)//
6(tan⁡2(x)+1)((tan⁡2(x)+1)2+7(tan⁡2(x)+1)tan⁡2(x)+2tan⁡4(x)−3tan⁡2(x)−1)6 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 7 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 2 \tan^{4}{\left(x \right)} - 3 \tan^{2}{\left(x \right)} - 1\right)
Simplify
The graph
Derivative of y=tg^(3)(x)-3tg(x)+3x
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