Mister Exam

Derivative of y=(x-1)²(x²-2x+3)³

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
                       3
       2 / 2          \ 
(x - 1) *\x  - 2*x + 3/ 
$$\left(x - 1\right)^{2} \left(\left(x^{2} - 2 x\right) + 3\right)^{3}$$
Detail solution
  1. Apply the product rule:

    ; to find :

    1. Let .

    2. Apply the power rule: goes to

    3. Then, apply the chain rule. Multiply by :

      1. Differentiate term by term:

        1. Apply the power rule: goes to

        2. The derivative of the constant is zero.

        The result is:

      The result of the chain rule is:

    ; to find :

    1. Let .

    2. Apply the power rule: goes to

    3. Then, apply the chain rule. Multiply by :

      1. Differentiate term by term:

        1. Differentiate term by term:

          1. Apply the power rule: goes to

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

            1. Apply the power rule: goes to

            So, the result is:

          The result is:

        2. The derivative of the constant is zero.

        The result is:

      The result of the chain rule is:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
              3                                     2           
/ 2          \                      2 / 2          \            
\x  - 2*x + 3/ *(-2 + 2*x) + (x - 1) *\x  - 2*x + 3/ *(-6 + 6*x)
$$\left(x - 1\right)^{2} \left(6 x - 6\right) \left(\left(x^{2} - 2 x\right) + 3\right)^{2} + \left(2 x - 2\right) \left(\left(x^{2} - 2 x\right) + 3\right)^{3}$$
The second derivative [src]
                 /              2                                                                         \
  /     2      \ |/     2      \              2 /     2                   2\              2 /     2      \|
2*\3 + x  - 2*x/*\\3 + x  - 2*x/  + 3*(-1 + x) *\3 + x  - 2*x + 4*(-1 + x) / + 12*(-1 + x) *\3 + x  - 2*x//
$$2 \left(x^{2} - 2 x + 3\right) \left(12 \left(x - 1\right)^{2} \left(x^{2} - 2 x + 3\right) + 3 \left(x - 1\right)^{2} \left(x^{2} - 2 x + 4 \left(x - 1\right)^{2} + 3\right) + \left(x^{2} - 2 x + 3\right)^{2}\right)$$
The third derivative [src]
            /                2                                                                                             \
            |  /     2      \              2 /                    2      2\     /     2      \ /     2                   2\|
12*(-1 + x)*\3*\3 + x  - 2*x/  + 2*(-1 + x) *\9 - 6*x + 2*(-1 + x)  + 3*x / + 3*\3 + x  - 2*x/*\3 + x  - 2*x + 4*(-1 + x) //
$$12 \left(x - 1\right) \left(2 \left(x - 1\right)^{2} \left(3 x^{2} - 6 x + 2 \left(x - 1\right)^{2} + 9\right) + 3 \left(x^{2} - 2 x + 3\right)^{2} + 3 \left(x^{2} - 2 x + 3\right) \left(x^{2} - 2 x + 4 \left(x - 1\right)^{2} + 3\right)\right)$$
The graph
Derivative of y=(x-1)²(x²-2x+3)³