Derivative of y=(x⁴-x²+1)³

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             3
/ 4    2    \ 
\x  - x  + 1/

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

(x^4 - x^2 + 1)^3

Detail solution

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

$3 \left(4 x^{3} - 2 x\right) \left(\left(x^{4} - x^{2}\right) + 1\right)^{2}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

             2               
/ 4    2    \  /           3\
\x  - x  + 1/ *\-6*x + 12*x /

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

Simplify

The second derivative [src]

  /                                            2\              
  |/        2\ /     4    2\      2 /        2\ | /     4    2\
6*\\-1 + 6*x /*\1 + x  - x / + 4*x *\-1 + 2*x / /*\1 + x  - x /

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

Simplify

The third derivative [src]

     /               2                   3                                          \
     |  /     4    2\       2 /        2\      /        2\ /        2\ /     4    2\|
24*x*\3*\1 + x  - x /  + 2*x *\-1 + 2*x /  + 3*\-1 + 2*x /*\-1 + 6*x /*\1 + x  - x //

$$24 x \left(2 x^{2} \left(2 x^{2} - 1\right)^{3} + 3 \left(2 x^{2} - 1\right) \left(6 x^{2} - 1\right) \left(x^{4} - x^{2} + 1\right) + 3 \left(x^{4} - x^{2} + 1\right)^{2}\right)$$

Simplify

The graph

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Derivative of y=(x⁴-x²+1)³

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The solution