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tg^6x
The derivative of the function/ tg^6x

Derivative of tg^6x

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

You have entered [src] 
   6   
tan (x)
tan⁡6(x)\tan^{6}{\left(x \right)} 
tan(x)^6
Detail solution

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

  2. Apply the power rule: u6u^{6} goes to 6u56 u^{5}

  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:

    6(sin⁡2(x)+cos⁡2(x))tan⁡5(x)cos⁡2(x)\frac{6 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{5}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

  4. Now simplify:

    6tan⁡5(x)cos⁡2(x)\frac{6 \tan^{5}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

The answer is:

6tan⁡5(x)cos⁡2(x)\frac{6 \tan^{5}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

The graph
02468-8-6-4-2-1010-200000000000200000000000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   5    /         2   \
tan (x)*\6 + 6*tan (x)/
(6tan⁡2(x)+6)tan⁡5(x)\left(6 \tan^{2}{\left(x \right)} + 6\right) \tan^{5}{\left(x \right)}
Simplify
The second derivative [src] 
     4    /       2   \ /         2   \
6*tan (x)*\1 + tan (x)/*\5 + 7*tan (x)/
6(tan⁡2(x)+1)(7tan⁡2(x)+5)tan⁡4(x)6 \left(\tan^{2}{\left(x \right)} + 1\right) \left(7 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)}
Simplify
The third derivative [src] 
                         /                         2                          \
      3    /       2   \ |   4        /       2   \         2    /       2   \|
24*tan (x)*\1 + tan (x)/*\tan (x) + 5*\1 + tan (x)/  + 8*tan (x)*\1 + tan (x)//
24(tan⁡2(x)+1)(5(tan⁡2(x)+1)2+8(tan⁡2(x)+1)tan⁡2(x)+tan⁡4(x))tan⁡3(x)24 \left(\tan^{2}{\left(x \right)} + 1\right) \left(5 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 8 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + \tan^{4}{\left(x \right)}\right) \tan^{3}{\left(x \right)}
Simplify
The graph
Derivative of tg^6x
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