Derivative of ysin(y)^2

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     2   
y*sin (y)

$$y \sin^{2}{\left(y \right)}$$

d /     2   \
--\y*sin (y)/
dy

$$\frac{d}{d y} y \sin^{2}{\left(y \right)}$$

Transform

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 $1$
$g{\left(y \right)} = \sin^{2}{\left(y \right)}$ ; to find $\frac{d}{d y} g{\left(y \right)}$ :
1. Let $u = \sin{\left(y \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d y} \sin{\left(y \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d y} \sin{\left(y \right)} = \cos{\left(y \right)}$
  The result of the chain rule is:
  
  $2 \sin{\left(y \right)} \cos{\left(y \right)}$
The result is: $2 y \sin{\left(y \right)} \cos{\left(y \right)} + \sin^{2}{\left(y \right)}$
Now simplify:

$y \sin{\left(2 y \right)} - \frac{\cos{\left(2 y \right)}}{2} + \frac{1}{2}$

The answer is:

$y \sin{\left(2 y \right)} - \frac{\cos{\left(2 y \right)}}{2} + \frac{1}{2}$

The graph

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The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /       2           2                       \
2*\- 3*sin (y) + 3*cos (y) - 4*y*cos(y)*sin(y)/

$$2 \left(- 4 y \sin{\left(y \right)} \cos{\left(y \right)} - 3 \sin^{2}{\left(y \right)} + 3 \cos^{2}{\left(y \right)}\right)$$

Simplify

The graph