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Identical expressions
sqrt(cos(7x- one))
square root of ( co sinus of e of (7x minus 1))
square root of ( co sinus of e of (7x minus one))
√(cos(7x-1))
sqrtcos7x-1
Similar expressions
sqrt(cos(7x+1))

The derivative of the function/ sqrt(cos(7x-1))

Derivative of sqrt(cos(7x-1))

The solution

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  ______________
\/ cos(7*x - 1)

$$\sqrt{\cos{\left(7 x - 1 \right)}}$$

d /  ______________\
--\\/ cos(7*x - 1) /
dx

$$\frac{d}{d x} \sqrt{\cos{\left(7 x - 1 \right)}}$$

Transform

Detail solution

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

$- \frac{7 \sin{\left(7 x - 1 \right)}}{2 \sqrt{\cos{\left(7 x - 1 \right)}}}$
Now simplify:

$- \frac{7 \sin{\left(7 x - 1 \right)}}{2 \sqrt{\cos{\left(7 x - 1 \right)}}}$

The answer is:

$- \frac{7 \sin{\left(7 x - 1 \right)}}{2 \sqrt{\cos{\left(7 x - 1 \right)}}}$

The graph

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Plot the graph f'(x)

The first derivative [src]

 -7*sin(7*x - 1)  
------------------
    ______________
2*\/ cos(7*x - 1)

$$- \frac{7 \sin{\left(7 x - 1 \right)}}{2 \sqrt{\cos{\left(7 x - 1 \right)}}}$$

Simplify

The second derivative [src]

    /                          2           \
    |    _______________    sin (-1 + 7*x) |
-49*|2*\/ cos(-1 + 7*x)  + ----------------|
    |                         3/2          |
    \                      cos   (-1 + 7*x)/
--------------------------------------------
                     4

$$- \frac{49 \left(\frac{\sin^{2}{\left(7 x - 1 \right)}}{\cos^{\frac{3}{2}}{\left(7 x - 1 \right)}} + 2 \sqrt{\cos{\left(7 x - 1 \right)}}\right)}{4}$$

Simplify

The third derivative [src]

     /         2          \              
     |    3*sin (-1 + 7*x)|              
-343*|2 + ----------------|*sin(-1 + 7*x)
     |        2           |              
     \     cos (-1 + 7*x) /              
-----------------------------------------
               _______________           
           8*\/ cos(-1 + 7*x)

$$- \frac{343 \cdot \left(\frac{3 \sin^{2}{\left(7 x - 1 \right)}}{\cos^{2}{\left(7 x - 1 \right)}} + 2\right) \sin{\left(7 x - 1 \right)}}{8 \sqrt{\cos{\left(7 x - 1 \right)}}}$$

Simplify

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Derivative of sqrt(cos(7x-1))

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The solution