Derivative of cos^-2(x)

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

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

cos(x)^(-2)

Detail solution

Let $u = \cos{\left(x \right)}$ .
Apply the power rule: $\frac{1}{u^{2}}$ goes to $- \frac{2}{u^{3}}$
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:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of cos^-2(x)

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The solution