Derivative of logcosxsinx

The solution

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log(cos(x))*x*sin(x)

$$x \log{\left(\cos{\left(x \right)} \right)} \sin{\left(x \right)}$$

(log(cos(x))*x)*sin(x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x \log{\left(\cos{\left(x \right)} \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \log{\left(\cos{\left(x \right)} \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = \cos{\left(x \right)}$ .
  2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
  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:
    
    $- \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
  $g{\left(x \right)} = x$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  The result is: $- \frac{x \sin{\left(x \right)}}{\cos{\left(x \right)}} + \log{\left(\cos{\left(x \right)} \right)}$
$g{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
The result is: $x \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)} + \left(- \frac{x \sin{\left(x \right)}}{\cos{\left(x \right)}} + \log{\left(\cos{\left(x \right)} \right)}\right) \sin{\left(x \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

/  x*sin(x)              \                              
|- -------- + log(cos(x))|*sin(x) + x*cos(x)*log(cos(x))
\   cos(x)               /

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

Simplify

The second derivative [src]

 //  /       2   \           \                                                                   \
 ||  |    sin (x)|   2*sin(x)|            /               x*sin(x)\                              |
-||x*|1 + -------| + --------|*sin(x) + 2*|-log(cos(x)) + --------|*cos(x) + x*log(cos(x))*sin(x)|
 ||  |       2   |    cos(x) |            \                cos(x) /                              |
 \\  \    cos (x)/           /                                                                   /

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

Simplify

The third derivative [src]

    /  /       2   \           \                                                                      /       2   \                        
    |  |    sin (x)|   2*sin(x)|            /               x*sin(x)\                                 |    sin (x)| /    2*x*sin(x)\       
- 3*|x*|1 + -------| + --------|*cos(x) + 3*|-log(cos(x)) + --------|*sin(x) - x*cos(x)*log(cos(x)) - |1 + -------|*|3 + ----------|*sin(x)
    |  |       2   |    cos(x) |            \                cos(x) /                                 |       2   | \      cos(x)  /       
    \  \    cos (x)/           /                                                                      \    cos (x)/

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

Simplify

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Derivative of logcosxsinx

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The solution