Mister Exam

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
The first derivative [src]
   /1\
sin|-|
   \x/
------
   2  
  x   
sin(1x)x2\frac{\sin{\left(\frac{1}{x} \right)}}{x^{2}}
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}}
The third derivative [src]
              /1\        /1\
           sin|-|   6*cos|-|
     /1\      \x/        \x/
6*sin|-| - ------ + --------
     \x/      2        x    
             x              
----------------------------
              4             
             x              
6sin(1x)+6cos(1x)xsin(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}}
The graph
Derivative of cos(1/x)