Mister Exam

Derivative of y=sqrt(cos3x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
  __________
\/ cos(3*x) 
$$\sqrt{\cos{\left(3 x \right)}}$$
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. 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:

      The result of the chain rule is:

    The result of the chain rule is:


The answer is:

The graph
The first derivative [src]
 -3*sin(3*x)  
--------------
    __________
2*\/ cos(3*x) 
$$- \frac{3 \sin{\left(3 x \right)}}{2 \sqrt{\cos{\left(3 x \right)}}}$$
The second derivative [src]
   /                     2      \
   |    __________    sin (3*x) |
-9*|2*\/ cos(3*x)  + -----------|
   |                    3/2     |
   \                 cos   (3*x)/
---------------------------------
                4                
$$- \frac{9 \left(\frac{\sin^{2}{\left(3 x \right)}}{\cos^{\frac{3}{2}}{\left(3 x \right)}} + 2 \sqrt{\cos{\left(3 x \right)}}\right)}{4}$$
The third derivative [src]
    /         2     \         
    |    3*sin (3*x)|         
-27*|2 + -----------|*sin(3*x)
    |        2      |         
    \     cos (3*x) /         
------------------------------
            __________        
        8*\/ cos(3*x)         
$$- \frac{27 \left(\frac{3 \sin^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} + 2\right) \sin{\left(3 x \right)}}{8 \sqrt{\cos{\left(3 x \right)}}}$$
The graph
Derivative of y=sqrt(cos3x)