Derivative of x/cos(x)

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  x   
------
cos(x)

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

x/cos(x)

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
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:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  /         2   \           
  |    2*sin (x)|   2*sin(x)
x*|1 + ---------| + --------
  |        2    |    cos(x) 
  \     cos (x) /           
----------------------------
           cos(x)

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of x/cos(x)

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Piecewise:

The solution