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(x^ three - three x+ two)^(two /3)
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Similar expressions
(x^3+3x+2)^(2/3)
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The derivative of the function/ (x^3-3x+2)^(2/3)

Derivative of (x^3-3x+2)^(2/3)

The solution

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

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

  /              2/3\
d |/ 3          \   |
--\\x  - 3*x + 2/   /
dx

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

Transform

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

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The first derivative [src]

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

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

Simplify

The second derivative [src]

  /                2 \
  |       /      2\  |
  |       \-1 + x /  |
2*|2*x - ------------|
  |           3      |
  \      2 + x  - 3*x/
----------------------
     ______________   
  3 /      3          
  \/  2 + x  - 3*x

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

Simplify

The third derivative [src]

  /                 3                 \
  |        /      2\         /      2\|
  |      2*\-1 + x /     3*x*\-1 + x /|
4*|1 + --------------- - -------------|
  |                  2         3      |
  |    /     3      \     2 + x  - 3*x|
  \    \2 + x  - 3*x/                 /
---------------------------------------
              ______________           
           3 /      3                  
           \/  2 + x  - 3*x

$$\frac{4 \left(- \frac{3 x \left(x^{2} - 1\right)}{x^{3} - 3 x + 2} + \frac{2 \left(x^{2} - 1\right)^{3}}{\left(x^{3} - 3 x + 2\right)^{2}} + 1\right)}{\sqrt[3]{x^{3} - 3 x + 2}}$$

Simplify

The graph

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Derivative of (x^3-3x+2)^(2/3)

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Piecewise:

The solution