Derivative of cos(1/x)

You have entered [src]

   /1\
cos|-|
   \x/

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

cos(1/x)

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of cos(1/x)

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