Mister Exam

Derivative of tg^6x

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
   6   
tan (x)
$$\tan^{6}{\left(x \right)}$$
tan(x)^6
Detail solution
  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:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
   5    /         2   \
tan (x)*\6 + 6*tan (x)/
$$\left(6 \tan^{2}{\left(x \right)} + 6\right) \tan^{5}{\left(x \right)}$$
The second derivative [src]
     4    /       2   \ /         2   \
6*tan (x)*\1 + tan (x)/*\5 + 7*tan (x)/
$$6 \left(\tan^{2}{\left(x \right)} + 1\right) \left(7 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)}$$
The third derivative [src]
                         /                         2                          \
      3    /       2   \ |   4        /       2   \         2    /       2   \|
24*tan (x)*\1 + tan (x)/*\tan (x) + 5*\1 + tan (x)/  + 8*tan (x)*\1 + tan (x)//
$$24 \left(\tan^{2}{\left(x \right)} + 1\right) \left(5 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 8 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + \tan^{4}{\left(x \right)}\right) \tan^{3}{\left(x \right)}$$
The graph
Derivative of tg^6x