Derivative of xlogcosx

The solution

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x*log(cos(x))*x

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

(x*log(cos(x)))*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:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = \log{\left(\cos{\left(x \right)} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  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:
    
    $- \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
  The result is: $- \frac{x \sin{\left(x \right)}}{\cos{\left(x \right)}} + \log{\left(\cos{\left(x \right)} \right)}$
$g{\left(x \right)} = x$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
The result is: $x \left(- \frac{x \sin{\left(x \right)}}{\cos{\left(x \right)}} + \log{\left(\cos{\left(x \right)} \right)}\right) + x \log{\left(\cos{\left(x \right)} \right)}$
Now simplify:

$x \left(- x \tan{\left(x \right)} + 2 \log{\left(\cos{\left(x \right)} \right)}\right)$

The answer is:

$x \left(- x \tan{\left(x \right)} + 2 \log{\left(\cos{\left(x \right)} \right)}\right)$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

 /    /       2   \                /       2   \                 \
 |    |    sin (x)|   6*sin(x)     |    sin (x)| /    2*x*sin(x)\|
-|3*x*|1 + -------| + -------- + x*|1 + -------|*|3 + ----------||
 |    |       2   |    cos(x)      |       2   | \      cos(x)  /|
 \    \    cos (x)/                \    cos (x)/                 /

$$- (x \left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \left(\frac{2 x \sin{\left(x \right)}}{\cos{\left(x \right)}} + 3\right) + 3 x \left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)}})$$

Simplify

The graph

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Identical expressions

Derivative of xlogcosx

The graph:

Piecewise:

The solution