Mister Exam

Lang:

The derivative of the function/ ycos(y)

Derivative of ycos(y)

The solution

You have entered [src]

y*cos(y)

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

y*cos(y)

Detail solution

Apply the product rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

-y*sin(y) + cos(y)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify