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

Derivative of tg^3x

Function f() - derivative -N order at the point
v

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The solution

You have entered [src] 
   3   
tan (x)
tan⁡3(x)\tan^{3}{\left(x \right)} 
d /   3   \
--\tan (x)/
dx
ddxtan⁡3(x)\frac{d}{d x} \tan^{3}{\left(x \right)}
Transform
Detail solution

  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. Now simplify:

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

The answer is:

3tan⁡2(x)cos⁡2(x)\frac{3 \tan^{2}{\left(x \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   \
tan (x)*\3 + 3*tan (x)/
(3tan⁡2(x)+3)tan⁡2(x)\left(3 \tan^{2}{\left(x \right)} + 3\right) \tan^{2}{\left(x \right)}
Simplify
The second derivative [src] 
  /       2   \ /         2   \       
6*\1 + tan (x)/*\1 + 2*tan (x)/*tan(x)
6(tan⁡2(x)+1)(2tan⁡2(x)+1)tan⁡(x)6 \left(\tan^{2}{\left(x \right)} + 1\right) \left(2 \tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}
Simplify
The third derivative [src] 
                /             2                                      \
  /       2   \ |/       2   \         4           2    /       2   \|
6*\1 + tan (x)/*\\1 + tan (x)/  + 2*tan (x) + 7*tan (x)*\1 + tan (x)//
6(tan⁡2(x)+1)(2tan⁡4(x)+7(tan⁡2(x)+1)tan⁡2(x)+(tan⁡2(x)+1)2)6 \left(\tan^{2}{\left(x \right)} + 1\right) \left(2 \tan^{4}{\left(x \right)} + 7 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right)^{2}\right)
Simplify
The graph
Derivative of tg^3x
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