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y=(x⁴-2x³+5x²-4)³
The derivative of the function/ y=(x⁴-2x³+5x²-4)³

Derivative of y=(x⁴-2x³+5x²-4)³

Function f() - derivative -N order at the point
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The solution

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                      3
/ 4      3      2    \ 
\x  - 2*x  + 5*x  - 4/
((5x2+(x4−2x3))−4)3\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

  1. Let u=(5x2+(x4−2x3))−4u = \left(5 x^{2} + \left(x^{4} - 2 x^{3}\right)\right) - 4.

  2. Apply the power rule: u3u^{3} goes to 3u23 u^{2}

  3. Then, apply the chain rule. Multiply by ddx((5x2+(x4−2x3))−4)\frac{d}{d x} \left(\left(5 x^{2} + \left(x^{4} - 2 x^{3}\right)\right) - 4\right):

    1. Differentiate (5x2+(x4−2x3))−4\left(5 x^{2} + \left(x^{4} - 2 x^{3}\right)\right) - 4 term by term:

      1. Differentiate 5x2+(x4−2x3)5 x^{2} + \left(x^{4} - 2 x^{3}\right) term by term:

        1. Differentiate x4−2x3x^{4} - 2 x^{3} term by term:

          1. Apply the power rule: x4x^{4} goes to 4x34 x^{3}

          2. The derivative of a constant times a function is the constant times the derivative of the function.

            1. Apply the power rule: x3x^{3} goes to 3x23 x^{2}

            So, the result is: −6x2- 6 x^{2}

          The result is: 4x3−6x24 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: x2x^{2} goes to 2x2 x

          So, the result is: 10x10 x

        The result is: 4x3−6x2+10x4 x^{3} - 6 x^{2} + 10 x

      2. The derivative of the constant −4-4 is zero.

      The result is: 4x3−6x2+10x4 x^{3} - 6 x^{2} + 10 x

    The result of the chain rule is:

    3((5x2+(x4−2x3))−4)2(4x3−6x2+10x)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)

  4. Now simplify:

    6x(2x2−3x+5)(x4−2x3+5x2−4)26 x \left(2 x^{2} - 3 x + 5\right) \left(x^{4} - 2 x^{3} + 5 x^{2} - 4\right)^{2}

The answer is:

6x(2x2−3x+5)(x4−2x3+5x2−4)26 x \left(2 x^{2} - 3 x + 5\right) \left(x^{4} - 2 x^{3} + 5 x^{2} - 4\right)^{2}

The graph
02468-8-6-4-2-1010-50000000000005000000000000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
                      2                         
/ 4      3      2    \  /      2       3       \
\x  - 2*x  + 5*x  - 4/ *\- 18*x  + 12*x  + 30*x/
((5x2+(x4−2x3))−4)2(12x3−18x2+30x)\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(4x2(2x2−3x+5)2+(6x2−6x+5)(x4−2x3+5x2−4))(x4−2x3+5x2−4)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(4x3(2x2−3x+5)3+6x(2x2−3x+5)(6x2−6x+5)(x4−2x3+5x2−4)+3(2x−1)(x4−2x3+5x2−4)2)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
The graph
Derivative of y=(x⁴-2x³+5x²-4)³
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