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Identical expressions
tanx^(one / three)
tangent of x to the power of (1 divide by 3)
tangent of x to the power of (one divide by three)
tanx(1/3)
tanx1/3
tanx^1/3
tanx^(1 divide by 3)

The derivative of the function/ tanx^(1/3)

Derivative of tanx^(1/3)

The solution

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3 ________
\/ tan(x)

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

tan(x)^(1/3)

Detail solution

Let $u = \tan{\left(x \right)}$ .
Apply the power rule: $\sqrt[3]{u}$ goes to $\frac{1}{3 u^{\frac{2}{3}}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(x \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = \cos{\left(x \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  Now plug in to the quotient rule:
  
  $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
The result of the chain rule is:

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

$\frac{1}{3 \cos^{2}{\left(x \right)} \tan^{\frac{2}{3}}{\left(x \right)}}$

The answer is:

$\frac{1}{3 \cos^{2}{\left(x \right)} \tan^{\frac{2}{3}}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2   
1   tan (x)
- + -------
3      3   
-----------
    2/3    
 tan   (x)

$$\frac{\frac{\tan^{2}{\left(x \right)}}{3} + \frac{1}{3}}{\tan^{\frac{2}{3}}{\left(x \right)}}$$

Simplify

The second derivative [src]

                /                      2   \
  /       2   \ |  3 ________   1 + tan (x)|
2*\1 + tan (x)/*|3*\/ tan(x)  - -----------|
                |                   5/3    |
                \                tan   (x) /
--------------------------------------------
                     9

$$\frac{2 \left(- \frac{\tan^{2}{\left(x \right)} + 1}{\tan^{\frac{5}{3}}{\left(x \right)}} + 3 \sqrt[3]{\tan{\left(x \right)}}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{9}$$

Simplify

The third derivative [src]

                /                                                2\
                |                 /       2   \     /       2   \ |
  /       2   \ |      4/3      9*\1 + tan (x)/   5*\1 + tan (x)/ |
2*\1 + tan (x)/*|18*tan   (x) - --------------- + ----------------|
                |                     2/3               8/3       |
                \                  tan   (x)         tan   (x)    /
-------------------------------------------------------------------
                                 27

$$\frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\frac{5 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{\frac{8}{3}}{\left(x \right)}} - \frac{9 \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan^{\frac{2}{3}}{\left(x \right)}} + 18 \tan^{\frac{4}{3}}{\left(x \right)}\right)}{27}$$

Simplify

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Identical expressions

Derivative of tanx^(1/3)

The graph:

Piecewise:

The solution