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y=(secx-tanx)(secx+tanx)
The derivative of the function/ y=(secx-tanx)(secx+tanx)

Derivative of y=(secx-tanx)(secx+tanx)

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

  1. Apply the product rule:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}

    f(x)=−tan⁡(x)+sec⁡(x)f{\left(x \right)} = - \tan{\left(x \right)} + \sec{\left(x \right)}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

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

      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)}}

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

        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)}}

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

    g(x)=tan⁡(x)+sec⁡(x)g{\left(x \right)} = \tan{\left(x \right)} + \sec{\left(x \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

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

      1. The derivative of secant is secant times tangent:

        ddxsec⁡(x)=tan⁡(x)sec⁡(x)\frac{d}{d x} \sec{\left(x \right)} = \tan{\left(x \right)} \sec{\left(x \right)}

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

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

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

  2. Now simplify:

    00

The answer is:

00

The graph
-0.010-0.008-0.006-0.004-0.0020.0100.0000.0020.0040.0060.0080.00
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
                  /       2                   \                     /        2                   \
(sec(x) - tan(x))*\1 + tan (x) + sec(x)*tan(x)/ + (sec(x) + tan(x))*\-1 - tan (x) + sec(x)*tan(x)/
(−tan⁡(x)+sec⁡(x))(tan⁡2(x)+tan⁡(x)sec⁡(x)+1)+(tan⁡(x)+sec⁡(x))(−tan⁡2(x)+tan⁡(x)sec⁡(x)−1)\left(- \tan{\left(x \right)} + \sec{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + \tan{\left(x \right)} \sec{\left(x \right)} + 1\right) + \left(\tan{\left(x \right)} + \sec{\left(x \right)}\right) \left(- \tan^{2}{\left(x \right)} + \tan{\left(x \right)} \sec{\left(x \right)} - 1\right)
Simplify
The second derivative [src] 
                  /   2             /       2   \            /       2   \       \                      /   2             /       2   \            /       2   \       \     /       2                   \ /       2                   \
(sec(x) + tan(x))*\tan (x)*sec(x) + \1 + tan (x)/*sec(x) - 2*\1 + tan (x)/*tan(x)/ - (-sec(x) + tan(x))*\tan (x)*sec(x) + \1 + tan (x)/*sec(x) + 2*\1 + tan (x)/*tan(x)/ - 2*\1 + tan (x) + sec(x)*tan(x)/*\1 + tan (x) - sec(x)*tan(x)/
−(tan⁡(x)−sec⁡(x))(2(tan⁡2(x)+1)tan⁡(x)+(tan⁡2(x)+1)sec⁡(x)+tan⁡2(x)sec⁡(x))+(tan⁡(x)+sec⁡(x))(−2(tan⁡2(x)+1)tan⁡(x)+(tan⁡2(x)+1)sec⁡(x)+tan⁡2(x)sec⁡(x))−2(tan⁡2(x)−tan⁡(x)sec⁡(x)+1)(tan⁡2(x)+tan⁡(x)sec⁡(x)+1)- \left(\tan{\left(x \right)} - \sec{\left(x \right)}\right) \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \sec{\left(x \right)} + \tan^{2}{\left(x \right)} \sec{\left(x \right)}\right) + \left(\tan{\left(x \right)} + \sec{\left(x \right)}\right) \left(- 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \sec{\left(x \right)} + \tan^{2}{\left(x \right)} \sec{\left(x \right)}\right) - 2 \left(\tan^{2}{\left(x \right)} - \tan{\left(x \right)} \sec{\left(x \right)} + 1\right) \left(\tan^{2}{\left(x \right)} + \tan{\left(x \right)} \sec{\left(x \right)} + 1\right)
Simplify
The third derivative [src] 
                     /               2                                                                           \                     /               2                                                                           \                                                                                                                                                                                                      
                     |  /       2   \       3                  2    /       2   \     /       2   \              |                     |  /       2   \       3                  2    /       2   \     /       2   \              |     /       2                   \ /   2             /       2   \            /       2   \       \     /       2                   \ /   2             /       2   \            /       2   \       \
- (-sec(x) + tan(x))*\2*\1 + tan (x)/  + tan (x)*sec(x) + 4*tan (x)*\1 + tan (x)/ + 5*\1 + tan (x)/*sec(x)*tan(x)/ - (sec(x) + tan(x))*\2*\1 + tan (x)/  - tan (x)*sec(x) + 4*tan (x)*\1 + tan (x)/ - 5*\1 + tan (x)/*sec(x)*tan(x)/ - 3*\1 + tan (x) - sec(x)*tan(x)/*\tan (x)*sec(x) + \1 + tan (x)/*sec(x) + 2*\1 + tan (x)/*tan(x)/ + 3*\1 + tan (x) + sec(x)*tan(x)/*\tan (x)*sec(x) + \1 + tan (x)/*sec(x) - 2*\1 + tan (x)/*tan(x)/
−(tan⁡(x)−sec⁡(x))(2(tan⁡2(x)+1)2+4(tan⁡2(x)+1)tan⁡2(x)+5(tan⁡2(x)+1)tan⁡(x)sec⁡(x)+tan⁡3(x)sec⁡(x))−(tan⁡(x)+sec⁡(x))(2(tan⁡2(x)+1)2+4(tan⁡2(x)+1)tan⁡2(x)−5(tan⁡2(x)+1)tan⁡(x)sec⁡(x)−tan⁡3(x)sec⁡(x))+3(−2(tan⁡2(x)+1)tan⁡(x)+(tan⁡2(x)+1)sec⁡(x)+tan⁡2(x)sec⁡(x))(tan⁡2(x)+tan⁡(x)sec⁡(x)+1)−3(2(tan⁡2(x)+1)tan⁡(x)+(tan⁡2(x)+1)sec⁡(x)+tan⁡2(x)sec⁡(x))(tan⁡2(x)−tan⁡(x)sec⁡(x)+1)- \left(\tan{\left(x \right)} - \sec{\left(x \right)}\right) \left(2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 5 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} \sec{\left(x \right)} + \tan^{3}{\left(x \right)} \sec{\left(x \right)}\right) - \left(\tan{\left(x \right)} + \sec{\left(x \right)}\right) \left(2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} - 5 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} \sec{\left(x \right)} - \tan^{3}{\left(x \right)} \sec{\left(x \right)}\right) + 3 \left(- 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \sec{\left(x \right)} + \tan^{2}{\left(x \right)} \sec{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + \tan{\left(x \right)} \sec{\left(x \right)} + 1\right) - 3 \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \sec{\left(x \right)} + \tan^{2}{\left(x \right)} \sec{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} - \tan{\left(x \right)} \sec{\left(x \right)} + 1\right)
Simplify
The graph
Derivative of y=(secx-tanx)(secx+tanx)
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