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Derivative of:
Derivative of 1/3*x^3
Derivative of x*x^(1/2)
Derivative of x^(5/2)
Derivative of (x-4)^2
Integral of d{x}:
1/cos^2(x)
Graphing y =:
1/cos^2(x)
Identical expressions
one /cos^ two (x)
1 divide by co sinus of e of squared (x)
one divide by co sinus of e of to the power of two (x)
1/cos2(x)
1/cos2x
1/cos²(x)
1/cos to the power of 2(x)
1/cos^2x
1 divide by cos^2(x)
Similar expressions
ln((1/cos^2x-1/2)^(1/2)+1/cosx)
1/cos^2x

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

Derivative of 1/cos^2(x)

The solution

You have entered [src]

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

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

1/(cos(x)^2)

Detail solution

Let $u = \cos^{2}{\left(x \right)}$ .
Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos^{2}{\left(x \right)}$ :
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:

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

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

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

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

The graph

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Identical expressions

Similar expressions

Derivative of 1/cos^2(x)

The graph:

Piecewise:

The solution