Derivative of cbrt(cos(x))

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3 ________
\/ cos(x)

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

cos(x)^(1/3)

Detail solution

Let $u = \cos{\left(x \right)}$ .
Apply the power rule: $\sqrt[3]{u}$ goes to $\frac{1}{3 u^{\frac{2}{3}}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$ :
1. The derivative of cosine is negative sine:
  
  $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
The result of the chain rule is:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  -sin(x)  
-----------
     2/3   
3*cos   (x)

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

Simplify

The second derivative [src]

 /                    2   \ 
 |  3 ________   2*sin (x)| 
-|3*\/ cos(x)  + ---------| 
 |                  5/3   | 
 \               cos   (x)/ 
----------------------------
             9

$$- \frac{\frac{2 \sin^{2}{\left(x \right)}}{\cos^{\frac{5}{3}}{\left(x \right)}} + 3 \sqrt[3]{\cos{\left(x \right)}}}{9}$$

Simplify

The third derivative [src]

 /          2   \        
 |    10*sin (x)|        
-|9 + ----------|*sin(x) 
 |        2     |        
 \     cos (x)  /        
-------------------------
             2/3         
       27*cos   (x)

$$- \frac{\left(\frac{10 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 9\right) \sin{\left(x \right)}}{27 \cos^{\frac{2}{3}}{\left(x \right)}}$$

Simplify

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Derivative of cbrt(cos(x))

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The solution