Derivative of y=(x⁴-2x³+5x²-4)³

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                      3
/ 4      3      2    \ 
\x  - 2*x  + 5*x  - 4/

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

(x^4 - 2*x^3 + 5*x^2 - 4)^3

Detail solution

Let $u = \left(5 x^{2} + \left(x^{4} - 2 x^{3}\right)\right) - 4$ .
Apply the power rule: $u^{3}$ goes to $3 u^{2}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\left(5 x^{2} + \left(x^{4} - 2 x^{3}\right)\right) - 4\right)$ :
1. Differentiate $\left(5 x^{2} + \left(x^{4} - 2 x^{3}\right)\right) - 4$ term by term:
  1. Differentiate $5 x^{2} + \left(x^{4} - 2 x^{3}\right)$ term by term:
    1. Differentiate $x^{4} - 2 x^{3}$ term by term:
      1. Apply the power rule: $x^{4}$ goes to $4 x^{3}$
      2. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x^{3}$ goes to $3 x^{2}$
        
        So, the result is: $- 6 x^{2}$
      The result is: $4 x^{3} - 6 x^{2}$
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x^{2}$ goes to $2 x$
      So, the result is: $10 x$
    The result is: $4 x^{3} - 6 x^{2} + 10 x$
  2. The derivative of the constant $-4$ is zero.
  The result is: $4 x^{3} - 6 x^{2} + 10 x$
The result of the chain rule is:

$3 \left(\left(5 x^{2} + \left(x^{4} - 2 x^{3}\right)\right) - 4\right)^{2} \left(4 x^{3} - 6 x^{2} + 10 x\right)$
Now simplify:

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

The answer is:

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

The graph

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The first derivative [src]

                      2                         
/ 4      3      2    \  /      2       3       \
\x  - 2*x  + 5*x  - 4/ *\- 18*x  + 12*x  + 30*x/

$$\left(\left(5 x^{2} + \left(x^{4} - 2 x^{3}\right)\right) - 4\right)^{2} \left(12 x^{3} - 18 x^{2} + 30 x\right)$$

Simplify

The second derivative [src]

  /                                                                2\                        
  |/             2\ /      4      3      2\      2 /             2\ | /      4      3      2\
6*\\5 - 6*x + 6*x /*\-4 + x  - 2*x  + 5*x / + 4*x *\5 - 3*x + 2*x / /*\-4 + x  - 2*x  + 5*x /

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

Simplify

The third derivative [src]

   /                         2                                   3                                                                \
   |  /      4      3      2\                  3 /             2\        /             2\ /             2\ /      4      3      2\|
12*\3*\-4 + x  - 2*x  + 5*x / *(-1 + 2*x) + 4*x *\5 - 3*x + 2*x /  + 6*x*\5 - 6*x + 6*x /*\5 - 3*x + 2*x /*\-4 + x  - 2*x  + 5*x //

$$12 \left(4 x^{3} \left(2 x^{2} - 3 x + 5\right)^{3} + 6 x \left(2 x^{2} - 3 x + 5\right) \left(6 x^{2} - 6 x + 5\right) \left(x^{4} - 2 x^{3} + 5 x^{2} - 4\right) + 3 \left(2 x - 1\right) \left(x^{4} - 2 x^{3} + 5 x^{2} - 4\right)^{2}\right)$$

Simplify

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Derivative of y=(x⁴-2x³+5x²-4)³

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The solution