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(cos^2*x) divide by log(x)

The derivative of the function/ (cos^2*x)/log(x)

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

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)/

$$\frac{d}{d x} \frac{\cos^{2}{\left(x \right)}}{\log{\left(x \right)}}$$

Transform

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \cos^{2}{\left(x \right)}$ and $g{\left(x \right)} = \log{\left(x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \cos{\left(x \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$ :
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  The result of the chain rule is:
  
  $- 2 \sin{\left(x \right)} \cos{\left(x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
Now plug in to the quotient rule:

$\frac{- 2 \log{\left(x \right)} \sin{\left(x \right)} \cos{\left(x \right)} - \frac{\cos^{2}{\left(x \right)}}{x}}{\log{\left(x \right)}^{2}}$
Now simplify:

$- \frac{x \log{\left(x \right)} \sin{\left(2 x \right)} + \frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}}{x \log{\left(x \right)}^{2}}$

The answer is:

$- \frac{x \log{\left(x \right)} \sin{\left(2 x \right)} + \frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}}{x \log{\left(x \right)}^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      2                      
   cos (x)    2*cos(x)*sin(x)
- --------- - ---------------
       2           log(x)    
  x*log (x)

$$- \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{\log{\left(x \right)}} - \frac{\cos^{2}{\left(x \right)}}{x \log{\left(x \right)}^{2}}$$

Simplify

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)

$$\frac{2 \sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)} + \frac{4 \sin{\left(x \right)} \cos{\left(x \right)}}{x \log{\left(x \right)}} + \frac{\left(1 + \frac{2}{\log{\left(x \right)}}\right) \cos^{2}{\left(x \right)}}{x^{2} \log{\left(x \right)}}}{\log{\left(x \right)}}$$

Simplify

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)

$$\frac{2 \cdot \left(4 \sin{\left(x \right)} \cos{\left(x \right)} - \frac{3 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right)}{x \log{\left(x \right)}} - \frac{3 \cdot \left(1 + \frac{2}{\log{\left(x \right)}}\right) \sin{\left(x \right)} \cos{\left(x \right)}}{x^{2} \log{\left(x \right)}} - \frac{\left(1 + \frac{3}{\log{\left(x \right)}} + \frac{3}{\log{\left(x \right)}^{2}}\right) \cos^{2}{\left(x \right)}}{x^{3} \log{\left(x \right)}}\right)}{\log{\left(x \right)}}$$

Simplify

The graph

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Identical expressions

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

The graph:

Piecewise:

The solution