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Derivative of:
Derivative of x^12
Derivative of -e^x
Derivative of e^(2-x)
Derivative of x^2-4*x
Identical expressions
((x+ one)*(x- one)^(two / three))
((x plus 1) multiply by (x minus 1) to the power of (2 divide by 3))
((x plus one) multiply by (x minus one) to the power of (two divide by three))
((x+1)*(x-1)(2/3))
x+1*x-12/3
((x+1)(x-1)^(2/3))
((x+1)(x-1)(2/3))
x+1x-12/3
x+1x-1^2/3
((x+1)*(x-1)^(2 divide by 3))
Similar expressions
((x+1)*(x+1)^(2/3))
((x-1)*(x-1)^(2/3))

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

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

The solution

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

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

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

Detail solution

Apply the product rule:

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

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

$\frac{5 x - 1}{3 \sqrt[3]{x - 1}}$

The answer is:

$\frac{5 x - 1}{3 \sqrt[3]{x - 1}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  /    1 + x \
2*|6 - ------|
  \    -1 + x/
--------------
   3 ________ 
 9*\/ -1 + x

$$\frac{2 \left(6 - \frac{x + 1}{x - 1}\right)}{9 \sqrt[3]{x - 1}}$$

Simplify

The third derivative [src]

  /     4*(1 + x)\
2*|-9 + ---------|
  \       -1 + x /
------------------
             4/3  
  27*(-1 + x)

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

Simplify