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(cos^2*x)/log(x)

Derivative of (cos^2*x)/log(x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
   2   
cos (x)
-------
 log(x)
$$\frac{\cos^{2}{\left(x \right)}}{\log{\left(x \right)}}$$
  /   2   \
d |cos (x)|
--|-------|
dx\ log(x)/
$$\frac{d}{d x} \frac{\cos^{2}{\left(x \right)}}{\log{\left(x \right)}}$$
Detail solution
  1. Apply the quotient rule, which is:

    and .

    To find :

    1. Let .

    2. Apply the power rule: goes to

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

      1. The derivative of cosine is negative sine:

      The result of the chain rule is:

    To find :

    1. The derivative of is .

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

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