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x-1/(cos(x))^2
The derivative of the function/ x-1/(cos(x))^2

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

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

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The solution

You have entered [src] 
         1   
x - 1*-------
         2   
      cos (x)
x−1⋅1cos⁡2(x)x - 1 \cdot \frac{1}{\cos^{2}{\left(x \right)}} 
d /         1   \
--|x - 1*-------|
dx|         2   |
  \      cos (x)/
ddx(x−1⋅1cos⁡2(x))\frac{d}{d x} \left(x - 1 \cdot \frac{1}{\cos^{2}{\left(x \right)}}\right)
Transform
Detail solution

  1. Differentiate x−1⋅1cos⁡2(x)x - 1 \cdot \frac{1}{\cos^{2}{\left(x \right)}} term by term:

    1. Apply the power rule: xx goes to 11

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

      1. Let u=cos⁡2(x)u = \cos^{2}{\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⁡2(x)\frac{d}{d x} \cos^{2}{\left(x \right)}:

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

        2. Apply the power rule: u2u^{2} goes to 2u2 u

        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:

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

        The result of the chain rule is:

        2sin⁡(x)cos⁡3(x)\frac{2 \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}}

      So, the result is: −2sin⁡(x)cos⁡3(x)- \frac{2 \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}}

    The result is: −2sin⁡(x)cos⁡3(x)+1- \frac{2 \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}} + 1

The answer is:

−2sin⁡(x)cos⁡3(x)+1- \frac{2 \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}} + 1

The graph
02468-8-6-4-2-1010-100000100000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
    2*sin(x)
1 - --------
       3    
    cos (x)
−2sin⁡(x)cos⁡3(x)+1- \frac{2 \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}} + 1
Simplify
The second derivative [src] 
   /         2   \
   |    3*sin (x)|
-2*|1 + ---------|
   |        2    |
   \     cos (x) /
------------------
        2         
     cos (x)
−2⋅(3sin⁡2(x)cos⁡2(x)+1)cos⁡2(x)- \frac{2 \cdot \left(\frac{3 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right)}{\cos^{2}{\left(x \right)}}
Simplify
The third derivative [src] 
   /         2   \       
   |    3*sin (x)|       
-8*|2 + ---------|*sin(x)
   |        2    |       
   \     cos (x) /       
-------------------------
            3            
         cos (x)
−8⋅(3sin⁡2(x)cos⁡2(x)+2)sin⁡(x)cos⁡3(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)}}
Simplify
The graph
Derivative of x-1/(cos(x))^2
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