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Identical expressions
(x^ three - two x^2+ five)^ eight
(x cubed minus 2x squared plus 5) to the power of 8
(x to the power of three minus two x squared plus five) to the power of eight
(x3-2x2+5)8
x3-2x2+58
(x³-2x²+5)⁸
(x to the power of 3-2x to the power of 2+5) to the power of 8
x^3-2x^2+5^8
Similar expressions
(x^3-2x^2-5)^8
(x^3+2x^2+5)^8

The derivative of the function/ (x^3-2x^2+5)^8

Derivative of (x^3-2x^2+5)^8

The solution

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               8
/ 3      2    \ 
\x  - 2*x  + 5/

$$\left(x^{3} - 2 x^{2} + 5\right)^{8}$$

  /               8\
d |/ 3      2    \ |
--\\x  - 2*x  + 5/ /
dx

$$\frac{d}{d x} \left(x^{3} - 2 x^{2} + 5\right)^{8}$$

Transform

Detail solution

Let $u = x^{3} - 2 x^{2} + 5$ .
Apply the power rule: $u^{8}$ goes to $8 u^{7}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{3} - 2 x^{2} + 5\right)$ :
1. Differentiate $x^{3} - 2 x^{2} + 5$ 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: $x^{2}$ goes to $2 x$
      So, the result is: $4 x$
    So, the result is: $- 4 x$
  3. The derivative of the constant $5$ is zero.
  The result is: $3 x^{2} - 4 x$
The result of the chain rule is:

$8 \cdot \left(3 x^{2} - 4 x\right) \left(x^{3} - 2 x^{2} + 5\right)^{7}$
Now simplify:

$8 x \left(3 x - 4\right) \left(x^{3} - 2 x^{2} + 5\right)^{7}$

The answer is:

$8 x \left(3 x - 4\right) \left(x^{3} - 2 x^{2} + 5\right)^{7}$

The graph

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Plot the graph f'(x)

The first derivative [src]

               7                
/ 3      2    \  /            2\
\x  - 2*x  + 5/ *\-32*x + 24*x /

$$\left(24 x^{2} - 32 x\right) \left(x^{3} - 2 x^{2} + 5\right)^{7}$$

Simplify

The second derivative [src]

                 6                                                  
  /     3      2\  /             /     3      2\      2           2\
8*\5 + x  - 2*x / *\2*(-2 + 3*x)*\5 + x  - 2*x / + 7*x *(-4 + 3*x) /

$$8 \cdot \left(7 x^{2} \left(3 x - 4\right)^{2} + 2 \cdot \left(3 x - 2\right) \left(x^{3} - 2 x^{2} + 5\right)\right) \left(x^{3} - 2 x^{2} + 5\right)^{6}$$

Simplify

The third derivative [src]

                  5 /               2                                                               \
   /     3      2\  |/     3      2\       3           3                             /     3      2\|
48*\5 + x  - 2*x / *\\5 + x  - 2*x /  + 7*x *(-4 + 3*x)  + 7*x*(-4 + 3*x)*(-2 + 3*x)*\5 + x  - 2*x //

$$48 \left(x^{3} - 2 x^{2} + 5\right)^{5} \cdot \left(7 x^{3} \left(3 x - 4\right)^{3} + 7 x \left(3 x - 4\right) \left(3 x - 2\right) \left(x^{3} - 2 x^{2} + 5\right) + \left(x^{3} - 2 x^{2} + 5\right)^{2}\right)$$

Simplify

The graph

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Derivative of (x^3-2x^2+5)^8

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The solution