Derivative of cox(x)*x

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cos(x)*x*x

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

(cos(x)*x)*x

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

x*(-x*sin(x) + cos(x)) + cos(x)*x

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

-6*sin(x) + x*(-3*cos(x) + x*sin(x)) - 3*x*cos(x)

$$x \left(x \sin{\left(x \right)} - 3 \cos{\left(x \right)}\right) - 3 x \cos{\left(x \right)} - 6 \sin{\left(x \right)}$$

Simplify