Mister Exam

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The derivative of the function/ x*cos(x)

Derivative of x*cos(x)

The solution

You have entered [src]

x*cos(x)

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

d           
--(x*cos(x))
dx

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

Transform

Detail solution

Apply the product rule:

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

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

The answer is:

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

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify