Mister Exam

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

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
   4         4   
sec (x) - tan (x)
$$- \tan^{4}{\left(x \right)} + \sec^{4}{\left(x \right)}$$
sec(x)^4 - tan(x)^4
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. Let .

      2. Apply the power rule: goes to

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

        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:

        The result of the chain rule is:

      So, the result is:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
     3    /         2   \        4          
- tan (x)*\4 + 4*tan (x)/ + 4*sec (x)*tan(x)
$$- \left(4 \tan^{2}{\left(x \right)} + 4\right) \tan^{3}{\left(x \right)} + 4 \tan{\left(x \right)} \sec^{4}{\left(x \right)}$$
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)/
$$4 \left(- 3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \tan^{2}{\left(x \right)} - 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{4}{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \sec^{4}{\left(x \right)} + 4 \tan^{2}{\left(x \right)} \sec^{4}{\left(x \right)}\right)$$
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)
$$8 \left(- 3 \left(\tan^{2}{\left(x \right)} + 1\right)^{3} - 10 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \tan^{2}{\left(x \right)} - 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{4}{\left(x \right)} + 7 \left(\tan^{2}{\left(x \right)} + 1\right) \sec^{4}{\left(x \right)} + 8 \tan^{2}{\left(x \right)} \sec^{4}{\left(x \right)}\right) \tan{\left(x \right)}$$
The graph
Derivative of y=sec^4x-tan^4x