Derivative of cosx/sinx

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cos(x)
------
sin(x)

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

cos(x)/sin(x)

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of cosine is negative sine:
  
  $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        2   
     cos (x)
-1 - -------
        2   
     sin (x)

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

Simplify

The second derivative [src]

/         2   \       
|    2*cos (x)|       
|2 + ---------|*cos(x)
|        2    |       
\     sin (x) /       
----------------------
        sin(x)

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

Simplify

The third derivative [src]

 /                        /         2   \\
 |                   2    |    6*cos (x)||
 |                cos (x)*|5 + ---------||
 |         2              |        2    ||
 |    3*cos (x)           \     sin (x) /|
-|2 + --------- + -----------------------|
 |        2                  2           |
 \     sin (x)            sin (x)        /

$$- (\frac{\left(5 + \frac{6 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 2 + \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}})$$

Simplify

The graph

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Derivative of cosx/sinx

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The solution