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(2x-x^ three)^ ten
(2x minus x cubed ) to the power of 10
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(2x-x to the power of 3) to the power of 10
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Similar expressions
(2x+x^3)^10

The derivative of the function/ (2x-x^3)^10

Derivative of (2x-x^3)^10

The solution

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          10
/       3\  
\2*x - x /

$$\left(- x^{3} + 2 x\right)^{10}$$

  /          10\
d |/       3\  |
--\\2*x - x /  /
dx

$$\frac{d}{d x} \left(- x^{3} + 2 x\right)^{10}$$

Transform

Detail solution

Let $u = - x^{3} + 2 x$ .
Apply the power rule: $u^{10}$ goes to $10 u^{9}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(- x^{3} + 2 x\right)$ :
1. Differentiate $- x^{3} + 2 x$ term by term:
  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: $2$
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
    So, the result is: $- 3 x^{2}$
  The result is: $2 - 3 x^{2}$
The result of the chain rule is:

$10 \cdot \left(2 - 3 x^{2}\right) \left(- x^{3} + 2 x\right)^{9}$
Now simplify:

$x^{9} \left(x^{2} - 2\right)^{9} \cdot \left(30 x^{2} - 20\right)$

The answer is:

$x^{9} \left(x^{2} - 2\right)^{9} \cdot \left(30 x^{2} - 20\right)$

The graph

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The first derivative [src]

          9             
/       3\  /         2\
\2*x - x / *\20 - 30*x /

$$\left(- 30 x^{2} + 20\right) \left(- x^{3} + 2 x\right)^{9}$$

Simplify

The second derivative [src]

               8 /             2                 \
    8 /      2\  |  /        2\       2 /      2\|
30*x *\-2 + x / *\3*\-2 + 3*x /  + 2*x *\-2 + x //

$$30 x^{8} \left(x^{2} - 2\right)^{8} \cdot \left(2 x^{2} \left(x^{2} - 2\right) + 3 \left(3 x^{2} - 2\right)^{2}\right)$$

Simplify

The third derivative [src]

               7 /              3               2                              \
    7 /      2\  |   /        2\     2 /      2\        2 /      2\ /        2\|
60*x *\-2 + x / *\12*\-2 + 3*x /  + x *\-2 + x /  + 27*x *\-2 + x /*\-2 + 3*x //

$$60 x^{7} \left(x^{2} - 2\right)^{7} \left(x^{2} \left(x^{2} - 2\right)^{2} + 27 x^{2} \left(x^{2} - 2\right) \left(3 x^{2} - 2\right) + 12 \left(3 x^{2} - 2\right)^{3}\right)$$

Simplify

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Derivative of (2x-x^3)^10

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The solution