Derivative of cos(3/x)

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   /3\
cos|-|
   \x/

$$\cos{\left(\frac{3}{x} \right)}$$

cos(3/x)

Detail solution

Let $u = \frac{3}{x}$ .
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 $\frac{d}{d x} \frac{3}{x}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $\frac{1}{x}$ goes to $- \frac{1}{x^{2}}$
  So, the result is: $- \frac{3}{x^{2}}$
The result of the chain rule is:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                /3\        /3\\
  |           3*sin|-|   6*cos|-||
  |     /3\        \x/        \x/|
9*|2*sin|-| - -------- + --------|
  |     \x/       2         x    |
  \              x               /
----------------------------------
                 4                
                x

$$\frac{9 \left(2 \sin{\left(\frac{3}{x} \right)} + \frac{6 \cos{\left(\frac{3}{x} \right)}}{x} - \frac{3 \sin{\left(\frac{3}{x} \right)}}{x^{2}}\right)}{x^{4}}$$

Simplify

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Derivative of cos(3/x)

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The solution