Mister Exam

Derivative of tanx^3

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
   3   
tan (x)
$$\tan^{3}{\left(x \right)}$$
d /   3   \
--\tan (x)/
dx         
$$\frac{d}{d x} \tan^{3}{\left(x \right)}$$
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]
   2    /         2   \
tan (x)*\3 + 3*tan (x)/
$$\left(3 \tan^{2}{\left(x \right)} + 3\right) \tan^{2}{\left(x \right)}$$
The second derivative [src]
  /       2   \ /         2   \       
6*\1 + tan (x)/*\1 + 2*tan (x)/*tan(x)
$$6 \left(\tan^{2}{\left(x \right)} + 1\right) \left(2 \tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}$$
The third derivative [src]
                /             2                                      \
  /       2   \ |/       2   \         4           2    /       2   \|
6*\1 + tan (x)/*\\1 + tan (x)/  + 2*tan (x) + 7*tan (x)*\1 + tan (x)//
$$6 \left(\tan^{2}{\left(x \right)} + 1\right) \left(2 \tan^{4}{\left(x \right)} + 7 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right)^{2}\right)$$
The graph
Derivative of tanx^3