Derivative of sqrt(cos(7x))

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  __________
\/ cos(7*x)

$$\sqrt{\cos{\left(7 x \right)}}$$

sqrt(cos(7*x))

Detail solution

Let $u = \cos{\left(7 x \right)}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(7 x \right)}$ :
1. Let $u = 7 x$ .
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 $\frac{d}{d x} 7 x$ :
  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: $7$
  The result of the chain rule is:
  
  $- 7 \sin{\left(7 x \right)}$
The result of the chain rule is:

$- \frac{7 \sin{\left(7 x \right)}}{2 \sqrt{\cos{\left(7 x \right)}}}$

The answer is:

$- \frac{7 \sin{\left(7 x \right)}}{2 \sqrt{\cos{\left(7 x \right)}}}$

The graph

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The first derivative [src]

 -7*sin(7*x)  
--------------
    __________
2*\/ cos(7*x)

$$- \frac{7 \sin{\left(7 x \right)}}{2 \sqrt{\cos{\left(7 x \right)}}}$$

Simplify

The second derivative [src]

    /                     2      \
    |    __________    sin (7*x) |
-49*|2*\/ cos(7*x)  + -----------|
    |                    3/2     |
    \                 cos   (7*x)/
----------------------------------
                4

$$- \frac{49 \left(\frac{\sin^{2}{\left(7 x \right)}}{\cos^{\frac{3}{2}}{\left(7 x \right)}} + 2 \sqrt{\cos{\left(7 x \right)}}\right)}{4}$$

Simplify

The third derivative [src]

     /         2     \         
     |    3*sin (7*x)|         
-343*|2 + -----------|*sin(7*x)
     |        2      |         
     \     cos (7*x) /         
-------------------------------
             __________        
         8*\/ cos(7*x)

$$- \frac{343 \left(\frac{3 \sin^{2}{\left(7 x \right)}}{\cos^{2}{\left(7 x \right)}} + 2\right) \sin{\left(7 x \right)}}{8 \sqrt{\cos{\left(7 x \right)}}}$$

Simplify

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Derivative of sqrt(cos(7x))

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The solution