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x^3*sin(1/x)

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

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

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//
$$\frac{d}{d x} x^{3} \sin{\left(1 \cdot \frac{1}{x} \right)}$$
Detail solution
  1. Apply the product rule:

    ; to find :

    1. Apply the power rule: goes to

    ; to find :

    1. Let .

    2. The derivative of sine is cosine:

    3. Then, apply the chain rule. Multiply by :

      1. Apply the quotient rule, which is:

        and .

        To find :

        1. The derivative of the constant is zero.

        To find :

        1. Apply the power rule: goes to

        Now plug in to the quotient rule:

      The result of the chain rule is:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
       /  1\      2    /  1\
- x*cos|1*-| + 3*x *sin|1*-|
       \  x/           \  x/
$$3 x^{2} \sin{\left(1 \cdot \frac{1}{x} \right)} - x \cos{\left(1 \cdot \frac{1}{x} \right)}$$
The second derivative [src]
                /1\             
             sin|-|             
       /1\      \x/          /1\
- 4*cos|-| - ------ + 6*x*sin|-|
       \x/     x             \x/
$$6 x \sin{\left(\frac{1}{x} \right)} - 4 \cos{\left(\frac{1}{x} \right)} - \frac{\sin{\left(\frac{1}{x} \right)}}{x}$$
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          
$$6 \sin{\left(\frac{1}{x} \right)} + \frac{9 \cdot \left(2 \cos{\left(\frac{1}{x} \right)} - \frac{\sin{\left(\frac{1}{x} \right)}}{x}\right)}{x} + \frac{- 6 \cos{\left(\frac{1}{x} \right)} + \frac{6 \sin{\left(\frac{1}{x} \right)}}{x} + \frac{\cos{\left(\frac{1}{x} \right)}}{x^{2}}}{x} - \frac{18 \cos{\left(\frac{1}{x} \right)}}{x}$$
The graph
Derivative of x^3*sin(1/x)