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Derivative of secx^2-2tanx

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

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Piecewise:

The solution

You have entered [src]
   2              
sec (x) - 2*tan(x)
$$- 2 \tan{\left(x \right)} + \sec^{2}{\left(x \right)}$$
sec(x)^2 - 2*tan(x)
Detail solution
  1. Differentiate term by term:

    1. Let .

    2. Apply the power rule: goes to

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

      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:

      The result of the chain rule is:

    4. 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:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
          2           2          
-2 - 2*tan (x) + 2*sec (x)*tan(x)
$$- 2 \tan^{2}{\left(x \right)} + 2 \tan{\left(x \right)} \sec^{2}{\left(x \right)} - 2$$
The second derivative [src]
  /   2    /       2   \     /       2   \               2       2   \
2*\sec (x)*\1 + tan (x)/ - 2*\1 + tan (x)/*tan(x) + 2*sec (x)*tan (x)/
$$2 \left(- 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \sec^{2}{\left(x \right)} + 2 \tan^{2}{\left(x \right)} \sec^{2}{\left(x \right)}\right)$$
The third derivative [src]
  /               2                                                                               \
  |  /       2   \         2    /       2   \        2       3           2    /       2   \       |
4*\- \1 + tan (x)/  - 2*tan (x)*\1 + tan (x)/ + 2*sec (x)*tan (x) + 4*sec (x)*\1 + tan (x)/*tan(x)/
$$4 \left(- \left(\tan^{2}{\left(x \right)} + 1\right)^{2} - 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} \sec^{2}{\left(x \right)} + 2 \tan^{3}{\left(x \right)} \sec^{2}{\left(x \right)}\right)$$