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(x^4-x-1)^4
The derivative of the function/ (x^4-x-1)^4

Derivative of (x^4-x-1)^4

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

You have entered [src] 
            4
/ 4        \ 
\x  - x - 1/
((x4−x)−1)4\left(\left(x^{4} - x\right) - 1\right)^{4} 
(x^4 - x - 1)^4
Detail solution

  1. Let u=(x4−x)−1u = \left(x^{4} - x\right) - 1.

  2. Apply the power rule: u4u^{4} goes to 4u34 u^{3}

  3. Then, apply the chain rule. Multiply by ddx((x4−x)−1)\frac{d}{d x} \left(\left(x^{4} - x\right) - 1\right):

    1. Differentiate (x4−x)−1\left(x^{4} - x\right) - 1 term by term:

      1. Differentiate x4−xx^{4} - x 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: xx goes to 11

          So, the result is: −1-1

        The result is: 4x3−14 x^{3} - 1

      2. The derivative of the constant −1-1 is zero.

      The result is: 4x3−14 x^{3} - 1

    The result of the chain rule is:

    4(4x3−1)((x4−x)−1)34 \left(4 x^{3} - 1\right) \left(\left(x^{4} - x\right) - 1\right)^{3}

  4. Now simplify:

    (4−16x3)(−x4+x+1)3\left(4 - 16 x^{3}\right) \left(- x^{4} + x + 1\right)^{3}

The answer is:

(4−16x3)(−x4+x+1)3\left(4 - 16 x^{3}\right) \left(- x^{4} + x + 1\right)^{3}

The graph
02468-8-6-4-2-1010-5000000000000000050000000000000000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
            3             
/ 4        \  /         3\
\x  - x - 1/ *\-4 + 16*x /
(16x3−4)((x4−x)−1)3\left(16 x^{3} - 4\right) \left(\left(x^{4} - x\right) - 1\right)^{3}
Simplify
The second derivative [src] 
               2 /           2                    \
   /         4\  |/        3\       2 /         4\|
12*\1 + x - x / *\\-1 + 4*x /  - 4*x *\1 + x - x //
12(−4x2(−x4+x+1)+(4x3−1)2)(−x4+x+1)212 \left(- 4 x^{2} \left(- x^{4} + x + 1\right) + \left(4 x^{3} - 1\right)^{2}\right) \left(- x^{4} + x + 1\right)^{2}
Simplify
The third derivative [src] 
                /             3                   2                                 \
   /         4\ |  /        3\        /         4\        2 /        3\ /         4\|
24*\1 + x - x /*\- \-1 + 4*x /  - 4*x*\1 + x - x /  + 18*x *\-1 + 4*x /*\1 + x - x //
24(−x4+x+1)(18x2(4x3−1)(−x4+x+1)−4x(−x4+x+1)2−(4x3−1)3)24 \left(- x^{4} + x + 1\right) \left(18 x^{2} \left(4 x^{3} - 1\right) \left(- x^{4} + x + 1\right) - 4 x \left(- x^{4} + x + 1\right)^{2} - \left(4 x^{3} - 1\right)^{3}\right)
Simplify
The graph
Derivative of (x^4-x-1)^4
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