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Identical expressions
x*cbrt(x)*sin(one /x)
x multiply by cubic root of (x) multiply by sinus of (1 divide by x)
x multiply by cubic root of (x) multiply by sinus of (one divide by x)
xcbrt(x)sin(1/x)
xcbrtxsin1/x
x*cbrt(x)*sin(1 divide by x)

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

Derivative of xcbrt(x)sin(1/x)

The solution

You have entered [src]

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

$$\sqrt[3]{x} x \sin{\left(\frac{1}{x} \right)}$$

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

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \sqrt[3]{x} x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = \sqrt[3]{x}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Apply the power rule: $\sqrt[3]{x}$ goes to $\frac{1}{3 x^{\frac{2}{3}}}$
  The result is: $\frac{4 \sqrt[3]{x}}{3}$
$g{\left(x \right)} = \sin{\left(\frac{1}{x} \right)}$ ; 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: $\frac{4 \sqrt[3]{x} \sin{\left(\frac{1}{x} \right)}}{3} - \frac{\cos{\left(\frac{1}{x} \right)}}{x^{\frac{2}{3}}}$
Now simplify:

$\frac{\frac{4 x \sin{\left(\frac{1}{x} \right)}}{3} - \cos{\left(\frac{1}{x} \right)}}{x^{\frac{2}{3}}}$

The answer is:

$\frac{\frac{4 x \sin{\left(\frac{1}{x} \right)}}{3} - \cos{\left(\frac{1}{x} \right)}}{x^{\frac{2}{3}}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     /1\     3 ___    /1\
  cos|-|   4*\/ x *sin|-|
     \x/              \x/
- ------ + --------------
    2/3          3       
   x

$$\frac{4 \sqrt[3]{x} \sin{\left(\frac{1}{x} \right)}}{3} - \frac{\cos{\left(\frac{1}{x} \right)}}{x^{\frac{2}{3}}}$$

Simplify

The second derivative [src]

                         /1\           
                      sin|-|           
     /1\        /1\      \x/        /1\
4*sin|-|   2*cos|-| - ------   8*cos|-|
     \x/        \x/     x           \x/
-------- + ----------------- - --------
   9               x             3*x   
---------------------------------------
                   2/3                 
                  x

$$\frac{\frac{4 \sin{\left(\frac{1}{x} \right)}}{9} + \frac{2 \cos{\left(\frac{1}{x} \right)} - \frac{\sin{\left(\frac{1}{x} \right)}}{x}}{x} - \frac{8 \cos{\left(\frac{1}{x} \right)}}{3 x}}{x^{\frac{2}{3}}}$$

Simplify

The third derivative [src]

                             /1\        /1\                                   
                          cos|-|   6*sin|-|     /              /1\\           
                    /1\      \x/        \x/     |           sin|-||           
       /1\   - 6*cos|-| + ------ + --------     |     /1\      \x/|        /1\
  8*sin|-|          \x/      2        x       4*|2*cos|-| - ------|   4*cos|-|
       \x/                  x                   \     \x/     x   /        \x/
- -------- + ------------------------------ + --------------------- - --------
     27                    x                            x               3*x   
------------------------------------------------------------------------------
                                      5/3                                     
                                     x

$$\frac{- \frac{8 \sin{\left(\frac{1}{x} \right)}}{27} + \frac{4 \left(2 \cos{\left(\frac{1}{x} \right)} - \frac{\sin{\left(\frac{1}{x} \right)}}{x}\right)}{x} + \frac{- 6 \cos{\left(\frac{1}{x} \right)} + \frac{6 \sin{\left(\frac{1}{x} \right)}}{x} + \frac{\cos{\left(\frac{1}{x} \right)}}{x^{2}}}{x} - \frac{4 \cos{\left(\frac{1}{x} \right)}}{3 x}}{x^{\frac{5}{3}}}$$

Simplify

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Identical expressions

Derivative of xcbrt(x)sin(1/x)

The graph:

Piecewise:

The solution

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How to use it?

Identical expressions

Derivative of x*cbrt(x)*sin(1/x)

The graph:

Piecewise:

The solution

Derivative of xcbrt(x)sin(1/x)