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Identical expressions
lncos(three x)^(one /3)
ln co sinus of e of (3x) to the power of (1 divide by 3)
ln co sinus of e of (three x) to the power of (one divide by 3)
lncos(3x)(1/3)
lncos3x1/3
lncos3x^1/3
lncos(3x)^(1 divide by 3)

The derivative of the function/ lncos(3x)^(1/3)

Derivative of lncos(3x)^(1/3)

The solution

You have entered [src]

3 _______________
\/ log(cos(3*x))

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

log(cos(3*x))^(1/3)

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

 /         2                   2           \ 
 |    3*sin (3*x)         2*sin (3*x)      | 
-|3 + ----------- + -----------------------| 
 |        2            2                   | 
 \     cos (3*x)    cos (3*x)*log(cos(3*x))/ 
---------------------------------------------
                  2/3                        
               log   (cos(3*x))

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of lncos(3x)^(1/3)

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The solution