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Identical expressions
cbrt(sqrt(cos(3x+ two)))
cubic root of ( square root of ( co sinus of e of (3x plus 2)))
cubic root of ( square root of ( co sinus of e of (3x plus two)))
cbrt(√(cos(3x+2)))
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Similar expressions
cbrt(sqrt(cos(3x-2)))

The derivative of the function/ cbrt(sqrt(cos(3x+2)))

Derivative of cbrt(sqrt(cos(3x+2)))

The solution

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   __________________
3 /   ______________ 
\/  \/ cos(3*x + 2)

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

(sqrt(cos(3*x + 2)))^(1/3)

Detail solution

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

$- \frac{\sin{\left(3 x + 2 \right)}}{2 \cos^{\frac{5}{6}}{\left(3 x + 2 \right)}}$
Now simplify:

$- \frac{\sin{\left(3 x + 2 \right)}}{2 \cos^{\frac{5}{6}}{\left(3 x + 2 \right)}}$

The answer is:

$- \frac{\sin{\left(3 x + 2 \right)}}{2 \cos^{\frac{5}{6}}{\left(3 x + 2 \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 6 ______________              
-\/ cos(3*x + 2) *sin(3*x + 2) 
-------------------------------
         2*cos(3*x + 2)

$$- \frac{\sin{\left(3 x + 2 \right)} \sqrt[6]{\cos{\left(3 x + 2 \right)}}}{2 \cos{\left(3 x + 2 \right)}}$$

Simplify

The second derivative [src]

 /                          2          \ 
 |  6 ______________   5*sin (2 + 3*x) | 
-|6*\/ cos(2 + 3*x)  + ----------------| 
 |                        11/6         | 
 \                     cos    (2 + 3*x)/ 
-----------------------------------------
                    4

$$- \frac{\frac{5 \sin^{2}{\left(3 x + 2 \right)}}{\cos^{\frac{11}{6}}{\left(3 x + 2 \right)}} + 6 \sqrt[6]{\cos{\left(3 x + 2 \right)}}}{4}$$

Simplify

The third derivative [src]

 /           2         \              
 |     55*sin (2 + 3*x)|              
-|54 + ----------------|*sin(2 + 3*x) 
 |         2           |              
 \      cos (2 + 3*x)  /              
--------------------------------------
               5/6                    
          8*cos   (2 + 3*x)

$$- \frac{\left(\frac{55 \sin^{2}{\left(3 x + 2 \right)}}{\cos^{2}{\left(3 x + 2 \right)}} + 54\right) \sin{\left(3 x + 2 \right)}}{8 \cos^{\frac{5}{6}}{\left(3 x + 2 \right)}}$$

Simplify

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Derivative of cbrt(sqrt(cos(3x+2)))

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The solution