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

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

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

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The solution

You have entered [src] 
 3    /  1\
x *sin|1*-|
      \  x/
x3sin⁡(1⋅1x)x^{3} \sin{\left(1 \cdot \frac{1}{x} \right)} 
d / 3    /  1\\
--|x *sin|1*-||
dx\      \  x//
ddxx3sin⁡(1⋅1x)\frac{d}{d x} x^{3} \sin{\left(1 \cdot \frac{1}{x} \right)}
Transform
Detail solution

  1. Apply the product rule:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\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(x)=x3f{\left(x \right)} = x^{3}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Apply the power rule: x3x^{3} goes to 3x23 x^{2}

    g(x)=sin⁡(1⋅1x)g{\left(x \right)} = \sin{\left(1 \cdot \frac{1}{x} \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Let u=1⋅1xu = 1 \cdot \frac{1}{x}.

    2. The derivative of sine is cosine:

      ddusin⁡(u)=cos⁡(u)\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}

    3. Then, apply the chain rule. Multiply by ddx1⋅1x\frac{d}{d x} 1 \cdot \frac{1}{x}:

      1. Apply the quotient rule, which is:

        ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=1f{\left(x \right)} = 1 and g(x)=xg{\left(x \right)} = x.

        To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

        1. The derivative of the constant 11 is zero.

        To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

        1. Apply the power rule: xx goes to 11

        Now plug in to the quotient rule:

        −1x2- \frac{1}{x^{2}}

      The result of the chain rule is:

      −cos⁡(1⋅1x)x2- \frac{\cos{\left(1 \cdot \frac{1}{x} \right)}}{x^{2}}

    The result is: 3x2sin⁡(1⋅1x)−xcos⁡(1⋅1x)3 x^{2} \sin{\left(1 \cdot \frac{1}{x} \right)} - x \cos{\left(1 \cdot \frac{1}{x} \right)}

  2. Now simplify:

    x(3xsin⁡(1x)−cos⁡(1x))x \left(3 x \sin{\left(\frac{1}{x} \right)} - \cos{\left(\frac{1}{x} \right)}\right)

The answer is:

x(3xsin⁡(1x)−cos⁡(1x))x \left(3 x \sin{\left(\frac{1}{x} \right)} - \cos{\left(\frac{1}{x} \right)}\right)

The graph
02468-8-6-4-2-1010200-100
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/
3x2sin⁡(1⋅1x)−xcos⁡(1⋅1x)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/
6xsin⁡(1x)−4cos⁡(1x)−sin⁡(1x)x6 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
6sin⁡(1x)+9⋅(2cos⁡(1x)−sin⁡(1x)x)x+−6cos⁡(1x)+6sin⁡(1x)x+cos⁡(1x)x2x−18cos⁡(1x)x6 \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
Derivative of x^3*sin(1/x)
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