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x- one /(cos(x))^ two
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x minus one divide by ( co sinus of e of (x)) to the power of two
x-1/(cos(x))2
x-1/cosx2
x-1/(cos(x))²
x-1/(cos(x)) to the power of 2
x-1/cosx^2
x-1 divide by (cos(x))^2
Similar expressions
x+1/(cos(x))^2
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The derivative of the function/ x-1/(cos(x))^2

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

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)/

$$\frac{d}{d x} \left(x - 1 \cdot \frac{1}{\cos^{2}{\left(x \right)}}\right)$$

Transform

Detail solution

Differentiate $x - 1 \cdot \frac{1}{\cos^{2}{\left(x \right)}}$ term by term:
1. Apply the power rule: $x$ goes to $1$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \cos^{2}{\left(x \right)}$ .
  2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos^{2}{\left(x \right)}$ :
    1. Let $u = \cos{\left(x \right)}$ .
    2. Apply the power rule: $u^{2}$ goes to $2 u$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$ :
      1. The derivative of cosine is negative sine:
        
        $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      The result of the chain rule is:
      
      $- 2 \sin{\left(x \right)} \cos{\left(x \right)}$
    The result of the chain rule is:
    
    $\frac{2 \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}}$
  So, the result is: $- \frac{2 \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}}$
The result is: $- \frac{2 \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}} + 1$

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    2*sin(x)
1 - --------
       3    
    cos (x)

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

Simplify

The second derivative [src]

   /         2   \
   |    3*sin (x)|
-2*|1 + ---------|
   |        2    |
   \     cos (x) /
------------------
        2         
     cos (x)

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

Simplify

The third derivative [src]

   /         2   \       
   |    3*sin (x)|       
-8*|2 + ---------|*sin(x)
   |        2    |       
   \     cos (x) /       
-------------------------
            3            
         cos (x)

$$- \frac{8 \cdot \left(\frac{3 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2\right) \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}}$$

Simplify

The graph

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

The graph:

Piecewise:

The solution