Derivative of 1/cos(2*x)

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

$$\frac{1}{\cos{\left(2 x \right)}}$$

1/cos(2*x)

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  /         2     \
  |    2*sin (2*x)|
4*|1 + -----------|
  |        2      |
  \     cos (2*x) /
-------------------
      cos(2*x)

$$\frac{4 \left(\frac{2 \sin^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}} + 1\right)}{\cos{\left(2 x \right)}}$$

Simplify

The third derivative [src]

  /         2     \         
  |    6*sin (2*x)|         
8*|5 + -----------|*sin(2*x)
  |        2      |         
  \     cos (2*x) /         
----------------------------
            2               
         cos (2*x)

$$\frac{8 \left(\frac{6 \sin^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}} + 5\right) \sin{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}}$$

Simplify

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Derivative of 1/cos(2*x)

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The solution