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The derivative of the function/ x^2sin(1/x)

Derivative of x^2sin(1/x)

The solution

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 2    /1\
x *sin|-|
      \x/

$$x^{2} \sin{\left(\frac{1}{x} \right)}$$

x^2*sin(1/x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x^{2}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{2}$ goes to $2 x$
$g{\left(x \right)} = \sin{\left(\frac{1}{x} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \frac{1}{x}$ .
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 $\frac{d}{d x} \frac{1}{x}$ :
  1. Apply the power rule: $\frac{1}{x}$ goes to $- \frac{1}{x^{2}}$
  The result of the chain rule is:
  
  $- \frac{\cos{\left(\frac{1}{x} \right)}}{x^{2}}$
The result is: $2 x \sin{\left(\frac{1}{x} \right)} - \cos{\left(\frac{1}{x} \right)}$

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

$$2 \sin{\left(\frac{1}{x} \right)} + \frac{2 \cos{\left(\frac{1}{x} \right)} - \frac{\sin{\left(\frac{1}{x} \right)}}{x}}{x} - \frac{4 \cos{\left(\frac{1}{x} \right)}}{x}$$

Simplify

The third derivative [src]

   /1\
cos|-|
   \x/
------
   4  
  x

$$\frac{\cos{\left(\frac{1}{x} \right)}}{x^{4}}$$

Simplify

The graph

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Identical expressions

Derivative of x^2sin(1/x)

The graph:

Piecewise:

The solution