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

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

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

The graph:

from to

Piecewise:

The solution

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

    1. Apply the power rule: goes to

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

      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:

      So, the result is:

    The result is:


The answer is:

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