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sqrt(cos(7x-1))

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

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  ______________
\/ cos(7*x - 1) 
$$\sqrt{\cos{\left(7 x - 1 \right)}}$$
d /  ______________\
--\\/ cos(7*x - 1) /
dx                  
$$\frac{d}{d x} \sqrt{\cos{\left(7 x - 1 \right)}}$$
Detail solution
  1. Let .

  2. Apply the power rule: goes to

  3. Then, apply the chain rule. Multiply by :

    1. Let .

    2. The derivative of cosine is negative sine:

    3. Then, apply the chain rule. Multiply by :

      1. Differentiate 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: goes to

          So, the result is:

        2. The derivative of the constant is zero.

        The result is:

      The result of the chain rule is:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
 -7*sin(7*x - 1)  
------------------
    ______________
2*\/ cos(7*x - 1) 
$$- \frac{7 \sin{\left(7 x - 1 \right)}}{2 \sqrt{\cos{\left(7 x - 1 \right)}}}$$
The second derivative [src]
    /                          2           \
    |    _______________    sin (-1 + 7*x) |
-49*|2*\/ cos(-1 + 7*x)  + ----------------|
    |                         3/2          |
    \                      cos   (-1 + 7*x)/
--------------------------------------------
                     4                      
$$- \frac{49 \left(\frac{\sin^{2}{\left(7 x - 1 \right)}}{\cos^{\frac{3}{2}}{\left(7 x - 1 \right)}} + 2 \sqrt{\cos{\left(7 x - 1 \right)}}\right)}{4}$$
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)            
$$- \frac{343 \cdot \left(\frac{3 \sin^{2}{\left(7 x - 1 \right)}}{\cos^{2}{\left(7 x - 1 \right)}} + 2\right) \sin{\left(7 x - 1 \right)}}{8 \sqrt{\cos{\left(7 x - 1 \right)}}}$$
The graph
Derivative of sqrt(cos(7x-1))