Mister Exam

Other calculators

Derivative of sin(x)/(log(x)+1)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
  sin(x)  
----------
log(x) + 1
$$\frac{\sin{\left(x \right)}}{\log{\left(x \right)} + 1}$$
sin(x)/(log(x) + 1)
Detail solution
  1. Apply the quotient rule, which is:

    and .

    To find :

    1. The derivative of sine is cosine:

    To find :

    1. Differentiate term by term:

      1. The derivative of the constant is zero.

      2. The derivative of is .

      The result is:

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

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