Derivative of (-1)/cos(x)

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 -1   
------
cos(x)

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

-1/cos(x)

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \cos{\left(x \right)}$ .
2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$ :
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  The result of the chain rule is:
  
  $\frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
So, the result is: $- \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of (-1)/cos(x)

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