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(-1)/cos(x)
The derivative of the function/ (-1)/cos(x)

Derivative of (-1)/cos(x)

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
 -1   
------
cos(x)
−1cos⁡(x)- \frac{1}{\cos{\left(x \right)}} 
-1/cos(x)
Detail solution

  1. The derivative of a constant times a function is the constant times the derivative of the function.

    1. Let u=cos⁡(x)u = \cos{\left(x \right)}.

    2. Apply the power rule: 1u\frac{1}{u} goes to −1u2- \frac{1}{u^{2}}

    3. Then, apply the chain rule. Multiply by ddxcos⁡(x)\frac{d}{d x} \cos{\left(x \right)}:

      1. The derivative of cosine is negative sine:

        ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

      The result of the chain rule is:

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

    So, the result is: −sin⁡(x)cos⁡2(x)- \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}

The answer is:

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

The graph
02468-8-6-4-2-1010-20002000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
-sin(x) 
--------
   2    
cos (x)
−sin⁡(x)cos⁡2(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)
−2sin⁡2(x)cos⁡2(x)+1cos⁡(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)
−(6sin⁡2(x)cos⁡2(x)+5)sin⁡(x)cos⁡2(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
The graph
Derivative of (-1)/cos(x)
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