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y=sec^4x-tan^4x
The derivative of the function/ y=sec^4x-tan^4x

Derivative of y=sec^4x-tan^4x

Function f() - derivative -N order at the point
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   4         4   
sec (x) - tan (x)
−tan⁡4(x)+sec⁡4(x)- \tan^{4}{\left(x \right)} + \sec^{4}{\left(x \right)} 
sec(x)^4 - tan(x)^4
Detail solution

  1. Differentiate −tan⁡4(x)+sec⁡4(x)- \tan^{4}{\left(x \right)} + \sec^{4}{\left(x \right)} term by term:

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

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

    3. Then, apply the chain rule. Multiply by ddxsec⁡(x)\frac{d}{d x} \sec{\left(x \right)}:

      1. Rewrite the function to be differentiated:

        sec⁡(x)=1cos⁡(x)\sec{\left(x \right)} = \frac{1}{\cos{\left(x \right)}}

      2. Let u=cos⁡(x)u = \cos{\left(x \right)}.

      3. Apply the power rule: 1u\frac{1}{u} goes to −1u2- \frac{1}{u^{2}}

      4. Then, apply the chain rule. Multiply by ddxcos⁡(x)\frac{d}{d x} \cos{\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)}

        The result of the chain rule is:

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

      The result of the chain rule is:

      4sin⁡(x)sec⁡3(x)cos⁡2(x)\frac{4 \sin{\left(x \right)} \sec^{3}{\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. 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: −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)}}

    The result is: −4(sin⁡2(x)+cos⁡2(x))tan⁡3(x)cos⁡2(x)+4sin⁡(x)sec⁡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)}} + \frac{4 \sin{\left(x \right)} \sec^{3}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

  2. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-200000200000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
     3    /         2   \        4          
- tan (x)*\4 + 4*tan (x)/ + 4*sec (x)*tan(x)
−(4tan⁡2(x)+4)tan⁡3(x)+4tan⁡(x)sec⁡4(x)- \left(4 \tan^{2}{\left(x \right)} + 4\right) \tan^{3}{\left(x \right)} + 4 \tan{\left(x \right)} \sec^{4}{\left(x \right)}
Simplify
The second derivative [src] 
  /                                       2                                                      \
  |   4    /       2   \     /       2   \     2           4    /       2   \        4       2   |
4*\sec (x)*\1 + tan (x)/ - 3*\1 + tan (x)/ *tan (x) - 2*tan (x)*\1 + tan (x)/ + 4*sec (x)*tan (x)/
4(−3(tan⁡2(x)+1)2tan⁡2(x)−2(tan⁡2(x)+1)tan⁡4(x)+(tan⁡2(x)+1)sec⁡4(x)+4tan⁡2(x)sec⁡4(x))4 \left(- 3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \tan^{2}{\left(x \right)} - 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{4}{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \sec^{4}{\left(x \right)} + 4 \tan^{2}{\left(x \right)} \sec^{4}{\left(x \right)}\right)
Simplify
The third derivative [src] 
  /                 3                   2                                                                                \       
  |    /       2   \       /       2   \     2           4    /       2   \        4    /       2   \        4       2   |       
8*\- 3*\1 + tan (x)/  - 10*\1 + tan (x)/ *tan (x) - 2*tan (x)*\1 + tan (x)/ + 7*sec (x)*\1 + tan (x)/ + 8*sec (x)*tan (x)/*tan(x)
8(−3(tan⁡2(x)+1)3−10(tan⁡2(x)+1)2tan⁡2(x)−2(tan⁡2(x)+1)tan⁡4(x)+7(tan⁡2(x)+1)sec⁡4(x)+8tan⁡2(x)sec⁡4(x))tan⁡(x)8 \left(- 3 \left(\tan^{2}{\left(x \right)} + 1\right)^{3} - 10 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \tan^{2}{\left(x \right)} - 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{4}{\left(x \right)} + 7 \left(\tan^{2}{\left(x \right)} + 1\right) \sec^{4}{\left(x \right)} + 8 \tan^{2}{\left(x \right)} \sec^{4}{\left(x \right)}\right) \tan{\left(x \right)}
Simplify
The graph
Derivative of y=sec^4x-tan^4x
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