Derivative of tan(5/x)

The solution

You have entered [src]

   /5\
tan|-|
   \x/

$$\tan{\left(\frac{5}{x} \right)}$$

tan(5/x)

Detail solution

Rewrite the function to be differentiated:

$\tan{\left(\frac{5}{x} \right)} = \frac{\sin{\left(\frac{5}{x} \right)}}{\cos{\left(\frac{5}{x} \right)}}$
Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \sin{\left(\frac{5}{x} \right)}$ and $g{\left(x \right)} = \cos{\left(\frac{5}{x} \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \frac{5}{x}$ .
2. The derivative of sine is cosine:
  
  $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{5}{x}$ :
  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: $- \frac{5}{x^{2}}$
  The result of the chain rule is:
  
  $- \frac{5 \cos{\left(\frac{5}{x} \right)}}{x^{2}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \frac{5}{x}$ .
2. The derivative of cosine is negative sine:
  
  $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{5}{x}$ :
  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: $- \frac{5}{x^{2}}$
  The result of the chain rule is:
  
  $\frac{5 \sin{\left(\frac{5}{x} \right)}}{x^{2}}$
Now plug in to the quotient rule:

$\frac{- \frac{5 \sin^{2}{\left(\frac{5}{x} \right)}}{x^{2}} - \frac{5 \cos^{2}{\left(\frac{5}{x} \right)}}{x^{2}}}{\cos^{2}{\left(\frac{5}{x} \right)}}$
Now simplify:

$- \frac{5}{x^{2} \cos^{2}{\left(\frac{5}{x} \right)}}$

The answer is:

$- \frac{5}{x^{2} \cos^{2}{\left(\frac{5}{x} \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   /       2/5\\
-5*|1 + tan |-||
   \        \x//
----------------
        2       
       x

$$- \frac{5 \left(\tan^{2}{\left(\frac{5}{x} \right)} + 1\right)}{x^{2}}$$

Simplify

The second derivative [src]

                 /         /5\\
                 |    5*tan|-||
   /       2/5\\ |         \x/|
10*|1 + tan |-||*|1 + --------|
   \        \x// \       x    /
-------------------------------
                3              
               x

$$\frac{10 \left(1 + \frac{5 \tan{\left(\frac{5}{x} \right)}}{x}\right) \left(\tan^{2}{\left(\frac{5}{x} \right)} + 1\right)}{x^{3}}$$

Simplify

The third derivative [src]

                  /       /       2/5\\         /5\         2/5\\
                  |    25*|1 + tan |-||   30*tan|-|   50*tan |-||
    /       2/5\\ |       \        \x//         \x/          \x/|
-10*|1 + tan |-||*|3 + ---------------- + --------- + ----------|
    \        \x// |            2              x            2    |
                  \           x                           x     /
-----------------------------------------------------------------
                                 4                               
                                x

$$- \frac{10 \left(\tan^{2}{\left(\frac{5}{x} \right)} + 1\right) \left(3 + \frac{30 \tan{\left(\frac{5}{x} \right)}}{x} + \frac{25 \left(\tan^{2}{\left(\frac{5}{x} \right)} + 1\right)}{x^{2}} + \frac{50 \tan^{2}{\left(\frac{5}{x} \right)}}{x^{2}}\right)}{x^{4}}$$

Simplify

The graph

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Identical expressions

Derivative of tan(5/x)

The graph:

Piecewise:

The solution