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

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

The graph:

from to

Piecewise:

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
  1. The derivative of a constant times a function is the constant times the derivative of the function.

    1. Let .

    2. The derivative of cosine is negative sine:

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

      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 power rule: goes to

          The result of the chain rule is:

        The result is:

      The result of the chain rule is:

    So, the result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
/     /1\         \              
|  cos|-|         |              
|     \x/      /1\|    /     /1\\
|- ------ + sin|-||*sin|x*sin|-||
\    x         \x//    \     \x//
$$\left(\sin{\left(\frac{1}{x} \right)} - \frac{\cos{\left(\frac{1}{x} \right)}}{x}\right) \sin{\left(x \sin{\left(\frac{1}{x} \right)} \right)}$$
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          
$$\left(\sin{\left(\frac{1}{x} \right)} - \frac{\cos{\left(\frac{1}{x} \right)}}{x}\right)^{2} \cos{\left(x \sin{\left(\frac{1}{x} \right)} \right)} - \frac{\sin{\left(\frac{1}{x} \right)} \sin{\left(x \sin{\left(\frac{1}{x} \right)} \right)}}{x^{3}}$$
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                     
$$- \left(\sin{\left(\frac{1}{x} \right)} - \frac{\cos{\left(\frac{1}{x} \right)}}{x}\right)^{3} \sin{\left(x \sin{\left(\frac{1}{x} \right)} \right)} - \frac{3 \left(\sin{\left(\frac{1}{x} \right)} - \frac{\cos{\left(\frac{1}{x} \right)}}{x}\right) \sin{\left(\frac{1}{x} \right)} \cos{\left(x \sin{\left(\frac{1}{x} \right)} \right)}}{x^{3}} + \frac{3 \sin{\left(\frac{1}{x} \right)} \sin{\left(x \sin{\left(\frac{1}{x} \right)} \right)}}{x^{4}} + \frac{\sin{\left(x \sin{\left(\frac{1}{x} \right)} \right)} \cos{\left(\frac{1}{x} \right)}}{x^{5}}$$