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Identical expressions
y*y*sin(y)^ three
y multiply by y multiply by sinus of (y) cubed
y multiply by y multiply by sinus of (y) to the power of three
y*y*sin(y)3
y*y*siny3
y*y*sin(y)³
y*y*sin(y) to the power of 3
yysin(y)^3
yysin(y)3
yysiny3
yysiny^3

The derivative of the function/ y*y*sin(y)^3

Derivative of yysin(y)^3

The solution

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       3   
y*y*sin (y)

$$y y \sin^{3}{\left(y \right)}$$

d /       3   \
--\y*y*sin (y)/
dy

$$\frac{d}{d y} y y \sin^{3}{\left(y \right)}$$

Transform

Detail solution

Apply the product rule:

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

$f{\left(y \right)} = y$ ; to find $\frac{d}{d y} f{\left(y \right)}$ :
1. Apply the power rule: $y$ goes to $1$
$g{\left(y \right)} = y$ ; to find $\frac{d}{d y} g{\left(y \right)}$ :
1. Apply the power rule: $y$ goes to $1$
$h{\left(y \right)} = \sin^{3}{\left(y \right)}$ ; to find $\frac{d}{d y} h{\left(y \right)}$ :
1. Let $u = \sin{\left(y \right)}$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
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:
  
  $3 \sin^{2}{\left(y \right)} \cos{\left(y \right)}$
The result is: $3 y^{2} \sin^{2}{\left(y \right)} \cos{\left(y \right)} + 2 y \sin^{3}{\left(y \right)}$
Now simplify:

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

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

       3         2    2          
2*y*sin (y) + 3*y *sin (y)*cos(y)

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

Simplify

The second derivative [src]

/     2         2 /   2           2   \                     \       
\2*sin (y) - 3*y *\sin (y) - 2*cos (y)/ + 12*y*cos(y)*sin(y)/*sin(y)

$$\left(- 3 y^{2} \left(\sin^{2}{\left(y \right)} - 2 \cos^{2}{\left(y \right)}\right) + 12 y \sin{\left(y \right)} \cos{\left(y \right)} + 2 \sin^{2}{\left(y \right)}\right) \sin{\left(y \right)}$$

Simplify

The third derivative [src]

  /     2              2 /       2           2   \              /   2           2   \       \
3*\6*sin (y)*cos(y) - y *\- 2*cos (y) + 7*sin (y)/*cos(y) - 6*y*\sin (y) - 2*cos (y)/*sin(y)/

$$3 \left(- y^{2} \cdot \left(7 \sin^{2}{\left(y \right)} - 2 \cos^{2}{\left(y \right)}\right) \cos{\left(y \right)} - 6 y \left(\sin^{2}{\left(y \right)} - 2 \cos^{2}{\left(y \right)}\right) \sin{\left(y \right)} + 6 \sin^{2}{\left(y \right)} \cos{\left(y \right)}\right)$$

Simplify

The graph

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Identical expressions

Derivative of yysin(y)^3

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Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Derivative of y*y*sin(y)^3

The graph:

Piecewise:

The solution

Derivative of yysin(y)^3