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Similar expressions
cos(x*(sin(1/x)))

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

Derivative of -cos(x*(sin(1/x)))

The solution

You have entered [src]

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

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

-cos(x*sin(1/x))

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = x \sin{\left(\frac{1}{x} \right)}$ .
2. The derivative of cosine is negative sine:
  
  $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} x \sin{\left(\frac{1}{x} \right)}$ :
  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 $1$
    $g{\left(x \right)} = \sin{\left(\frac{1}{x} \right)}$ ; 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: $\sin{\left(\frac{1}{x} \right)} - \frac{\cos{\left(\frac{1}{x} \right)}}{x}$
  The result of the chain rule is:
  
  $- \left(\sin{\left(\frac{1}{x} \right)} - \frac{\cos{\left(\frac{1}{x} \right)}}{x}\right) \sin{\left(x \sin{\left(\frac{1}{x} \right)} \right)}$
So, the result is: $\left(\sin{\left(\frac{1}{x} \right)} - \frac{\cos{\left(\frac{1}{x} \right)}}{x}\right) \sin{\left(x \sin{\left(\frac{1}{x} \right)} \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                                                                                         /     /1\         \                     
                     3                                                                   |  cos|-|         |                     
  /     /1\         \                     /1\    /     /1\\        /1\    /     /1\\     |     \x/      /1\|    /     /1\\    /1\
  |  cos|-|         |                  cos|-|*sin|x*sin|-||   3*sin|-|*sin|x*sin|-||   3*|- ------ + sin|-||*cos|x*sin|-||*sin|-|
  |     \x/      /1\|     /     /1\\      \x/    \     \x//        \x/    \     \x//     \    x         \x//    \     \x//    \x/
- |- ------ + sin|-|| *sin|x*sin|-|| + -------------------- + ---------------------- - ------------------------------------------
  \    x         \x//     \     \x//             5                       4                                  3                    
                                                x                       x                                  x

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

Simplify

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Derivative of -cos(x*(sin(1/x)))

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The solution