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Limit of the function:
x^3*sin(1/x)
Identical expressions
x^ three *sin(one /x)
x cubed multiply by sinus of (1 divide by x)
x to the power of three multiply by sinus of (one divide by x)
x3*sin(1/x)
x3*sin1/x
x³*sin(1/x)
x to the power of 3*sin(1/x)
x^3sin(1/x)
x3sin(1/x)
x3sin1/x
x^3sin1/x
x^3*sin(1 divide by x)

The derivative of the function/ x^3*sin(1/x)

Derivative of x^3*sin(1/x)

The solution

You have entered [src]

 3    /  1\
x *sin|1*-|
      \  x/

$$x^{3} \sin{\left(1 \cdot \frac{1}{x} \right)}$$

d / 3    /  1\\
--|x *sin|1*-||
dx\      \  x//

$$\frac{d}{d x} x^{3} \sin{\left(1 \cdot \frac{1}{x} \right)}$$

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x^{3}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
$g{\left(x \right)} = \sin{\left(1 \cdot \frac{1}{x} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 1 \cdot \frac{1}{x}$ .
2. The derivative of sine is cosine:
  
  $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 1 \cdot \frac{1}{x}$ :
  1. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = 1$ and $g{\left(x \right)} = x$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. The derivative of the constant $1$ is zero.
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Apply the power rule: $x$ goes to $1$
    Now plug in to the quotient rule:
    
    $- \frac{1}{x^{2}}$
  The result of the chain rule is:
  
  $- \frac{\cos{\left(1 \cdot \frac{1}{x} \right)}}{x^{2}}$
The result is: $3 x^{2} \sin{\left(1 \cdot \frac{1}{x} \right)} - x \cos{\left(1 \cdot \frac{1}{x} \right)}$
Now simplify:

$x \left(3 x \sin{\left(\frac{1}{x} \right)} - \cos{\left(\frac{1}{x} \right)}\right)$

The answer is:

$x \left(3 x \sin{\left(\frac{1}{x} \right)} - \cos{\left(\frac{1}{x} \right)}\right)$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       /  1\      2    /  1\
- x*cos|1*-| + 3*x *sin|1*-|
       \  x/           \  x/

$$3 x^{2} \sin{\left(1 \cdot \frac{1}{x} \right)} - x \cos{\left(1 \cdot \frac{1}{x} \right)}$$

Simplify

The second derivative [src]

                /1\             
             sin|-|             
       /1\      \x/          /1\
- 4*cos|-| - ------ + 6*x*sin|-|
       \x/     x             \x/

$$6 x \sin{\left(\frac{1}{x} \right)} - 4 \cos{\left(\frac{1}{x} \right)} - \frac{\sin{\left(\frac{1}{x} \right)}}{x}$$

Simplify

The third derivative [src]

                           /1\        /1\                                    
                        cos|-|   6*sin|-|                 /              /1\\
                  /1\      \x/        \x/                 |           sin|-||
           - 6*cos|-| + ------ + --------         /1\     |     /1\      \x/|
                  \x/      2        x       18*cos|-|   9*|2*cos|-| - ------|
     /1\                  x                       \x/     \     \x/     x   /
6*sin|-| + ------------------------------ - --------- + ---------------------
     \x/                 x                      x                 x

$$6 \sin{\left(\frac{1}{x} \right)} + \frac{9 \cdot \left(2 \cos{\left(\frac{1}{x} \right)} - \frac{\sin{\left(\frac{1}{x} \right)}}{x}\right)}{x} + \frac{- 6 \cos{\left(\frac{1}{x} \right)} + \frac{6 \sin{\left(\frac{1}{x} \right)}}{x} + \frac{\cos{\left(\frac{1}{x} \right)}}{x^{2}}}{x} - \frac{18 \cos{\left(\frac{1}{x} \right)}}{x}$$

Simplify

The graph

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Identical expressions

Derivative of x^3*sin(1/x)

The graph:

Piecewise:

The solution