Mister Exam

Derivative of (ln(tg(x)+ctg(x)))/(sin(a))

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
log(tan(x) + cot(x))
--------------------
       sin(a)       
$$\frac{\log{\left(\tan{\left(x \right)} + \cot{\left(x \right)} \right)}}{\sin{\left(a \right)}}$$
log(tan(x) + cot(x))/sin(a)
Detail solution
  1. The derivative of a constant times a function is the constant times the derivative of the function.

    1. Let .

    2. The derivative of is .

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

      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. 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:

      The result of the chain rule is:

    So, the result is:

  2. Now simplify:


The answer is:

The first derivative [src]
      2         2       
   tan (x) - cot (x)    
------------------------
(tan(x) + cot(x))*sin(a)
$$\frac{\tan^{2}{\left(x \right)} - \cot^{2}{\left(x \right)}}{\left(\tan{\left(x \right)} + \cot{\left(x \right)}\right) \sin{\left(a \right)}}$$
The second derivative [src]
                     2                                                  
  /   2         2   \                                                   
  \tan (x) - cot (x)/      /       2   \            /       2   \       
- -------------------- + 2*\1 + cot (x)/*cot(x) + 2*\1 + tan (x)/*tan(x)
    cot(x) + tan(x)                                                     
------------------------------------------------------------------------
                        (cot(x) + tan(x))*sin(a)                        
$$\frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 2 \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} - \frac{\left(\tan^{2}{\left(x \right)} - \cot^{2}{\left(x \right)}\right)^{2}}{\tan{\left(x \right)} + \cot{\left(x \right)}}}{\left(\tan{\left(x \right)} + \cot{\left(x \right)}\right) \sin{\left(a \right)}}$$
The third derivative [src]
   /                                                     3                                                                                                                          \
   |             2                2   /   2         2   \                                                          /   2         2   \ //       2   \          /       2   \       \|
   |/       2   \    /       2   \    \tan (x) - cot (x)/         2    /       2   \        2    /       2   \   3*\tan (x) - cot (x)/*\\1 + cot (x)/*cot(x) + \1 + tan (x)/*tan(x)/|
-2*|\1 + cot (x)/  - \1 + tan (x)/  - -------------------- - 2*tan (x)*\1 + tan (x)/ + 2*cot (x)*\1 + cot (x)/ + -------------------------------------------------------------------|
   |                                                    2                                                                                  cot(x) + tan(x)                          |
   \                                   (cot(x) + tan(x))                                                                                                                            /
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                               (cot(x) + tan(x))*sin(a)                                                                              
$$- \frac{2 \left(\frac{3 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} - \cot^{2}{\left(x \right)}\right)}{\tan{\left(x \right)} + \cot{\left(x \right)}} - \left(\tan^{2}{\left(x \right)} + 1\right)^{2} - 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + \left(\cot^{2}{\left(x \right)} + 1\right)^{2} + 2 \left(\cot^{2}{\left(x \right)} + 1\right) \cot^{2}{\left(x \right)} - \frac{\left(\tan^{2}{\left(x \right)} - \cot^{2}{\left(x \right)}\right)^{3}}{\left(\tan{\left(x \right)} + \cot{\left(x \right)}\right)^{2}}\right)}{\left(\tan{\left(x \right)} + \cot{\left(x \right)}\right) \sin{\left(a \right)}}$$