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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. 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)}}

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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