Mister Exam

Other calculators

Derivative of ln(x)/sin(x)+x*ctg(x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
log(x)           
------ + x*cot(x)
sin(x)           
$$x \cot{\left(x \right)} + \frac{\log{\left(x \right)}}{\sin{\left(x \right)}}$$
log(x)/sin(x) + x*cot(x)
Detail solution
  1. Differentiate term by term:

    1. Apply the quotient rule, which is:

      and .

      To find :

      1. The derivative of is .

      To find :

      1. The derivative of sine is cosine:

      Now plug in to the quotient rule:

    2. Apply the product rule:

      ; to find :

      1. Apply the power rule: goes to

      ; to find :

      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 :

          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:

        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 is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
  /        2   \      1       cos(x)*log(x)         
x*\-1 - cot (x)/ + -------- - ------------- + cot(x)
                   x*sin(x)         2               
                                 sin (x)            
$$x \left(- \cot^{2}{\left(x \right)} - 1\right) - \frac{\log{\left(x \right)} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \cot{\left(x \right)} + \frac{1}{x \sin{\left(x \right)}}$$
The second derivative [src]
                                                                                  2          
          2      log(x)       1        2*cos(x)       /       2   \          2*cos (x)*log(x)
-2 - 2*cot (x) + ------ - --------- - --------- + 2*x*\1 + cot (x)/*cot(x) + ----------------
                 sin(x)    2               2                                        3        
                          x *sin(x)   x*sin (x)                                  sin (x)     
$$2 x \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + \frac{\log{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2 \log{\left(x \right)} \cos^{2}{\left(x \right)}}{\sin^{3}{\left(x \right)}} - 2 \cot^{2}{\left(x \right)} - 2 - \frac{2 \cos{\left(x \right)}}{x \sin^{2}{\left(x \right)}} - \frac{1}{x^{2} \sin{\left(x \right)}}$$
The third derivative [src]
                   2                                                        3                                                                             2   
      /       2   \        2          3         /       2   \          6*cos (x)*log(x)   5*cos(x)*log(x)          2    /       2   \    3*cos(x)    6*cos (x)
- 2*x*\1 + cot (x)/  + --------- + -------- + 6*\1 + cot (x)/*cot(x) - ---------------- - --------------- - 4*x*cot (x)*\1 + cot (x)/ + ---------- + ---------
                        3          x*sin(x)                                   4                  2                                       2    2           3   
                       x *sin(x)                                           sin (x)            sin (x)                                   x *sin (x)   x*sin (x)
$$- 2 x \left(\cot^{2}{\left(x \right)} + 1\right)^{2} - 4 x \left(\cot^{2}{\left(x \right)} + 1\right) \cot^{2}{\left(x \right)} + 6 \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} - \frac{5 \log{\left(x \right)} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{6 \log{\left(x \right)} \cos^{3}{\left(x \right)}}{\sin^{4}{\left(x \right)}} + \frac{3}{x \sin{\left(x \right)}} + \frac{6 \cos^{2}{\left(x \right)}}{x \sin^{3}{\left(x \right)}} + \frac{3 \cos{\left(x \right)}}{x^{2} \sin^{2}{\left(x \right)}} + \frac{2}{x^{3} \sin{\left(x \right)}}$$