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Derivative of:
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Derivative of 7x
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Identical expressions
(cos(two *x))^(- zero . five)
( co sinus of e of (2 multiply by x)) to the power of ( minus 0.5)
( co sinus of e of (two multiply by x)) to the power of ( minus zero . five)
(cos(2*x))(-0.5)
cos2*x-0.5
(cos(2x))^(-0.5)
(cos(2x))(-0.5)
cos2x-0.5
cos2x^-0.5
Similar expressions
(cos(2*x))^(0.5)

The derivative of the function/ (cos(2*x))^(-0.5)

Derivative of (cos(2*x))^(-0.5)

The solution

You have entered [src]

     1      
------------
  __________
\/ cos(2*x)

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

d /     1      \
--|------------|
dx|  __________|
  \\/ cos(2*x) /

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

Transform

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

/           2     \         
|     15*sin (2*x)|         
|14 + ------------|*sin(2*x)
|         2       |         
\      cos (2*x)  /         
----------------------------
           3/2              
        cos   (2*x)

$$\frac{\left(\frac{15 \sin^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}} + 14\right) \sin{\left(2 x \right)}}{\cos^{\frac{3}{2}}{\left(2 x \right)}}$$

Simplify

The graph

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Derivative of (cos(2*x))^(-0.5)

The graph:

Piecewise:

The solution