Mister Exam

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

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

The graph:

from to

Piecewise:

The solution

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(sec(x) - tan(x))*(sec(x) + tan(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:

    ; to find :

    1. Differentiate term by term:

      1. Rewrite the function to be differentiated:

      2. Let .

      3. Apply the power rule: goes to

      4. Then, apply the chain rule. Multiply by :

        1. The derivative of cosine is negative sine:

        The result of the chain rule is:

      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:

        2. Apply the quotient rule, which is:

          and .

          To find :

          1. The derivative of sine is cosine:

          To find :

          1. The derivative of cosine is negative sine:

          Now plug in to the quotient rule:

        So, the result is:

      The result is:

    ; to find :

    1. Differentiate term by term:

      1. The derivative of secant is secant times tangent:

      The result is:

    The result is:

  2. Now simplify:


The answer is:

The graph
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)/
$$\left(- \tan{\left(x \right)} + \sec{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + \tan{\left(x \right)} \sec{\left(x \right)} + 1\right) + \left(\tan{\left(x \right)} + \sec{\left(x \right)}\right) \left(- \tan^{2}{\left(x \right)} + \tan{\left(x \right)} \sec{\left(x \right)} - 1\right)$$
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)/
$$- \left(\tan{\left(x \right)} - \sec{\left(x \right)}\right) \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \sec{\left(x \right)} + \tan^{2}{\left(x \right)} \sec{\left(x \right)}\right) + \left(\tan{\left(x \right)} + \sec{\left(x \right)}\right) \left(- 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \sec{\left(x \right)} + \tan^{2}{\left(x \right)} \sec{\left(x \right)}\right) - 2 \left(\tan^{2}{\left(x \right)} - \tan{\left(x \right)} \sec{\left(x \right)} + 1\right) \left(\tan^{2}{\left(x \right)} + \tan{\left(x \right)} \sec{\left(x \right)} + 1\right)$$
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)/
$$- \left(\tan{\left(x \right)} - \sec{\left(x \right)}\right) \left(2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 5 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} \sec{\left(x \right)} + \tan^{3}{\left(x \right)} \sec{\left(x \right)}\right) - \left(\tan{\left(x \right)} + \sec{\left(x \right)}\right) \left(2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} - 5 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} \sec{\left(x \right)} - \tan^{3}{\left(x \right)} \sec{\left(x \right)}\right) + 3 \left(- 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \sec{\left(x \right)} + \tan^{2}{\left(x \right)} \sec{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + \tan{\left(x \right)} \sec{\left(x \right)} + 1\right) - 3 \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \sec{\left(x \right)} + \tan^{2}{\left(x \right)} \sec{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} - \tan{\left(x \right)} \sec{\left(x \right)} + 1\right)$$
The graph
Derivative of y=(secx-tanx)(secx+tanx)