Derivative of sinx/cosx

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

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

sin(x)/cos(x)

Detail solution

Apply the quotient rule, which is:

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

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of sinx/cosx

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The solution