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Derivative of x^(1/3)/(3*x+2)
Identical expressions
x^(one / three)/(three *x+ two)
x to the power of (1 divide by 3) divide by (3 multiply by x plus 2)
x to the power of (one divide by three) divide by (three multiply by x plus two)
x(1/3)/(3*x+2)
x1/3/3*x+2
x^(1/3)/(3x+2)
x(1/3)/(3x+2)
x1/3/3x+2
x^1/3/3x+2
x^(1 divide by 3) divide by (3*x+2)
Similar expressions
x^(1/3)/(3*x-2)

The derivative of the function/ x^(1/3)/(3*x+2)

Derivative of x^(1/3)/(3*x+2)

The solution

You have entered [src]

 3 ___ 
 \/ x  
-------
3*x + 2

$$\frac{\sqrt[3]{x}}{3 x + 2}$$

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

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \sqrt[3]{x}$ and $g{\left(x \right)} = 3 x + 2$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $\sqrt[3]{x}$ goes to $\frac{1}{3 x^{\frac{2}{3}}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $3 x + 2$ term by term:
  1. The derivative of the constant $2$ is zero.
  2. 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 is: $3$
Now plug in to the quotient rule:

$\frac{- 3 \sqrt[3]{x} + \frac{3 x + 2}{3 x^{\frac{2}{3}}}}{\left(3 x + 2\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     3 ___                     
   3*\/ x             1        
- ---------- + ----------------
           2      2/3          
  (3*x + 2)    3*x   *(3*x + 2)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                               3 ___                   \
  |   5            1           81*\/ x            9       |
2*|------- + -------------- - ---------- + ---------------|
  |    8/3    5/3                      3    2/3          2|
  \27*x      x   *(2 + 3*x)   (2 + 3*x)    x   *(2 + 3*x) /
-----------------------------------------------------------
                          2 + 3*x

$$\frac{2 \left(- \frac{81 \sqrt[3]{x}}{\left(3 x + 2\right)^{3}} + \frac{9}{x^{\frac{2}{3}} \left(3 x + 2\right)^{2}} + \frac{1}{x^{\frac{5}{3}} \left(3 x + 2\right)} + \frac{5}{27 x^{\frac{8}{3}}}\right)}{3 x + 2}$$

Simplify

The graph

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Derivative of x^(1/3)/(3*x+2)

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The solution