Mister Exam

Other calculators

cos(1/x)
The derivative of the function/ cos(1/x)

Derivative of cos(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/
cos⁡(1x)\cos{\left(\frac{1}{x} \right)} 
cos(1/x)
Detail solution

  1. Let u=1xu = \frac{1}{x}.

  2. The derivative of cosine is negative sine:

    dducos⁡(u)=−sin⁡(u)\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}

  3. Then, apply the chain rule. Multiply by ddx1x\frac{d}{d x} \frac{1}{x}:

    1. Apply the power rule: 1x\frac{1}{x} goes to −1x2- \frac{1}{x^{2}}

    The result of the chain rule is:

    sin⁡(1x)x2\frac{\sin{\left(\frac{1}{x} \right)}}{x^{2}}

The answer is:

sin⁡(1x)x2\frac{\sin{\left(\frac{1}{x} \right)}}{x^{2}}

The graph
02468-8-6-4-2-1010-100100
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   /1\
sin|-|
   \x/
------
   2  
  x
sin⁡(1x)x2\frac{\sin{\left(\frac{1}{x} \right)}}{x^{2}}
Simplify
The second derivative [src] 
 /              /1\\ 
 |           cos|-|| 
 |     /1\      \x/| 
-|2*sin|-| + ------| 
 \     \x/     x   / 
---------------------
           3         
          x
−2sin⁡(1x)+cos⁡(1x)xx3- \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
6sin⁡(1x)+6cos⁡(1x)x−sin⁡(1x)x2x4\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
The graph
Derivative of cos(1/x)
    © Mister Exam – Calculator