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

Derivative of (x-2)^2*(x-4)+5

Function f() - derivative -N order at the point
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       2            
(x - 2) *(x - 4) + 5
(x4)(x2)2+5\left(x - 4\right) \left(x - 2\right)^{2} + 5
(x - 2)^2*(x - 4) + 5
Detail solution
  1. Differentiate (x4)(x2)2+5\left(x - 4\right) \left(x - 2\right)^{2} + 5 term by term:

    1. Apply the product rule:

      ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}

      f(x)=(x2)2f{\left(x \right)} = \left(x - 2\right)^{2}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. Let u=x2u = x - 2.

      2. Apply the power rule: u2u^{2} goes to 2u2 u

      3. Then, apply the chain rule. Multiply by ddx(x2)\frac{d}{d x} \left(x - 2\right):

        1. Differentiate x2x - 2 term by term:

          1. Apply the power rule: xx goes to 11

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

          The result is: 11

        The result of the chain rule is:

        2x42 x - 4

      g(x)=x4g{\left(x \right)} = x - 4; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. Differentiate x4x - 4 term by term:

        1. Apply the power rule: xx goes to 11

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

        The result is: 11

      The result is: (x4)(2x4)+(x2)2\left(x - 4\right) \left(2 x - 4\right) + \left(x - 2\right)^{2}

    2. The derivative of the constant 55 is zero.

    The result is: (x4)(2x4)+(x2)2\left(x - 4\right) \left(2 x - 4\right) + \left(x - 2\right)^{2}

  2. Now simplify:

    (x2)(3x10)\left(x - 2\right) \left(3 x - 10\right)


The answer is:

(x2)(3x10)\left(x - 2\right) \left(3 x - 10\right)

The graph
02468-8-6-4-2-1010-25002500
The first derivative [src]
       2                     
(x - 2)  + (-4 + 2*x)*(x - 4)
(x4)(2x4)+(x2)2\left(x - 4\right) \left(2 x - 4\right) + \left(x - 2\right)^{2}
The second derivative [src]
2*(-8 + 3*x)
2(3x8)2 \left(3 x - 8\right)
The third derivative [src]
6
66
The graph
Derivative of (x-2)^2*(x-4)+5