Mister Exam

Derivative of 2*tg(x)+ctg(x)

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

The graph:

from to

Piecewise:

The solution

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

    1. 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:

    2. There are multiple ways to do this derivative.

      Method #1

      1. Rewrite the function to be differentiated:

      2. Let .

      3. Apply the power rule: goes to

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

        The result of the chain rule is:

      Method #2

      1. Rewrite the function to be differentiated:

      2. Apply the quotient rule, which is:

        and .

        To find :

        1. The derivative of cosine is negative sine:

        To find :

        1. The derivative of sine is cosine:

        Now plug in to the quotient rule:

    The result is:

  2. Now simplify:


The answer is:

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