Mister Exam

Lang:

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

Derivative of x*sin(x)

The solution

You have entered [src]

x*sin(x)

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

x*sin(x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
$g{\left(x \right)} = \sin{\left(x \right)}$ ; 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: $x \cos{\left(x \right)} + \sin{\left(x \right)}$

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

x*cos(x) + sin(x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Derivative of x*sin(x)