导数 sqrt(2cos3x)

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  ____________
\/ 2*cos(3*x)

$$\sqrt{2 \cos{\left(3 x \right)}}$$

sqrt(2*cos(3*x))

Detail solution

Let $u = 2 \cos{\left(3 x \right)}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 \cos{\left(3 x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = 3 x$ .
  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 $\frac{d}{d x} 3 x$ :
    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: $3$
    The result of the chain rule is:
    
    $- 3 \sin{\left(3 x \right)}$
  So, the result is: $- 6 \sin{\left(3 x \right)}$
The result of the chain rule is:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

         /                     2      \
     ___ |    __________    sin (3*x) |
-9*\/ 2 *|2*\/ cos(3*x)  + -----------|
         |                    3/2     |
         \                 cos   (3*x)/
---------------------------------------
                   4

$$- \frac{9 \sqrt{2} \left(\frac{\sin^{2}{\left(3 x \right)}}{\cos^{\frac{3}{2}}{\left(3 x \right)}} + 2 \sqrt{\cos{\left(3 x \right)}}\right)}{4}$$

Simplify

The third derivative [src]

          /         2     \         
      ___ |    3*sin (3*x)|         
-27*\/ 2 *|2 + -----------|*sin(3*x)
          |        2      |         
          \     cos (3*x) /         
------------------------------------
               __________           
           8*\/ cos(3*x)

$$- \frac{27 \sqrt{2} \left(\frac{3 \sin^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} + 2\right) \sin{\left(3 x \right)}}{8 \sqrt{\cos{\left(3 x \right)}}}$$

Simplify

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导数 sqrt(2cos3x)

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