Mister Exam

Other calculators

Derivative of 1/(cos^2x)-1

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
   1       
------- - 1
   2       
cos (x)    
$$-1 + \frac{1}{\cos^{2}{\left(x \right)}}$$
1/(cos(x)^2) - 1
Detail solution
  1. Differentiate term by term:

    1. Let .

    2. Apply the power rule: goes to

    3. Then, apply the chain rule. Multiply by :

      1. Let .

      2. Apply the power rule: goes to

      3. Then, apply the chain rule. Multiply by :

        1. The derivative of cosine is negative sine:

        The result of the chain rule is:

      The result of the chain rule is:

    4. The derivative of the constant is zero.

    The result is:


The answer is:

The graph
The first derivative [src]
   2*sin(x)   
--------------
          2   
cos(x)*cos (x)
$$\frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)} \cos^{2}{\left(x \right)}}$$
The second derivative [src]
  /         2   \
  |    3*sin (x)|
2*|1 + ---------|
  |        2    |
  \     cos (x) /
-----------------
        2        
     cos (x)     
$$\frac{2 \left(\frac{3 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right)}{\cos^{2}{\left(x \right)}}$$
The third derivative [src]
  /         2   \       
  |    3*sin (x)|       
8*|2 + ---------|*sin(x)
  |        2    |       
  \     cos (x) /       
------------------------
           3            
        cos (x)         
$$\frac{8 \left(\frac{3 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2\right) \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}}$$