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The derivative of the function/ (1/4)*tan(x)^(4)

Derivative of (1/4)*tan(x)^(4)

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

You have entered [src] 
   4   
tan (x)
-------
   4
tan⁡4(x)4\frac{\tan^{4}{\left(x \right)}}{4} 
tan(x)^4/4
Detail solution

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

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

    2. Apply the power rule: u4u^{4} goes to 4u34 u^{3}

    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:

      4(sin⁡2(x)+cos⁡2(x))tan⁡3(x)cos⁡2(x)\frac{4 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{3}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

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

  2. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-5000000050000000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   3    /         2   \
tan (x)*\4 + 4*tan (x)/
-----------------------
           4
(4tan⁡2(x)+4)tan⁡3(x)4\frac{\left(4 \tan^{2}{\left(x \right)} + 4\right) \tan^{3}{\left(x \right)}}{4}
Simplify
The second derivative [src] 
   2    /       2   \ /         2   \
tan (x)*\1 + tan (x)/*\3 + 5*tan (x)/
(tan⁡2(x)+1)(5tan⁡2(x)+3)tan⁡2(x)\left(\tan^{2}{\left(x \right)} + 1\right) \left(5 \tan^{2}{\left(x \right)} + 3\right) \tan^{2}{\left(x \right)}
Simplify
The third derivative [src] 
                /                           2                           \       
  /       2   \ |     4        /       2   \          2    /       2   \|       
2*\1 + tan (x)/*\2*tan (x) + 3*\1 + tan (x)/  + 10*tan (x)*\1 + tan (x)//*tan(x)
2(tan⁡2(x)+1)(3(tan⁡2(x)+1)2+10(tan⁡2(x)+1)tan⁡2(x)+2tan⁡4(x))tan⁡(x)2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 10 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 2 \tan^{4}{\left(x \right)}\right) \tan{\left(x \right)}
Simplify
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