Mister Exam

Derivative of tgx-ctgx

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

The graph:

from to

Piecewise:

The solution

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

    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:

    3. The derivative of a constant times a function is the constant times the derivative of the function.

      1. 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:

      So, the result is:

    The result is:

  2. Now simplify:


The answer is:

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