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tgx+tg^3x
The derivative of the function/ tgx+tg^3x

Derivative of tgx+tg^3x

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

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            3   
tan(x) + tan (x)
tan⁡3(x)+tan⁡(x)\tan^{3}{\left(x \right)} + \tan{\left(x \right)} 
d /            3   \
--\tan(x) + tan (x)/
dx
ddx(tan⁡3(x)+tan⁡(x))\frac{d}{d x} \left(\tan^{3}{\left(x \right)} + \tan{\left(x \right)}\right)
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Detail solution

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

    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)}}

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

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

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

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

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

  2. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-10104000000-2000000
Plot the graph f(x)
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The first derivative [src] 
       2         2    /         2   \
1 + tan (x) + tan (x)*\3 + 3*tan (x)/
(3tan⁡2(x)+3)tan⁡2(x)+tan⁡2(x)+1\left(3 \tan^{2}{\left(x \right)} + 3\right) \tan^{2}{\left(x \right)} + \tan^{2}{\left(x \right)} + 1
Simplify
The second derivative [src] 
  /       2   \ /         2   \       
2*\1 + tan (x)/*\4 + 6*tan (x)/*tan(x)
2(tan⁡2(x)+1)(6tan⁡2(x)+4)tan⁡(x)2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(6 \tan^{2}{\left(x \right)} + 4\right) \tan{\left(x \right)}
Simplify
The third derivative [src] 
                /                   2                                                   \
  /       2   \ |      /       2   \         2           4            2    /       2   \|
2*\1 + tan (x)/*\1 + 3*\1 + tan (x)/  + 3*tan (x) + 6*tan (x) + 21*tan (x)*\1 + tan (x)//
2(tan⁡2(x)+1)(6tan⁡4(x)+21(tan⁡2(x)+1)tan⁡2(x)+3(tan⁡2(x)+1)2+3tan⁡2(x)+1)2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(6 \tan^{4}{\left(x \right)} + 21 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 3 \tan^{2}{\left(x \right)} + 1\right)
Simplify
The graph
Derivative of tgx+tg^3x
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