Derivative of sin(1/x)

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   /1\
sin|-|
   \x/

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

sin(1/x)

Detail solution

Let $u = \frac{1}{x}$ .
The derivative of sine is cosine:

$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
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 answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    /1\ 
-cos|-| 
    \x/ 
--------
    2   
   x

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                /1\        /1\
             cos|-|   6*sin|-|
       /1\      \x/        \x/
- 6*cos|-| + ------ + --------
       \x/      2        x    
               x              
------------------------------
               4              
              x

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

Simplify

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Derivative of sin(1/x)

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