Find the first and second derivatives of the function:  w = (z+1) (z-1) (z^2+1)

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to notice first that you may convert the product `(z+1)(z-1)`  into a difference of squares such that:

`(z+1)(z-1) = z^2 - 1`

Hence, you may write the transformed form of the equation of the function such that:

`w(z) = (z^2 - 1)(z^2+ 1)`

You need to convert again the product into a difference of squares such that:

`w(z) = z^4 - 1`

You need to differentiate the function with respect to z such that:

`w'(z) = 4z^3 - 0`

`w'(z) = 4z^3`

You need to find the second order derivative, hence, you need to differentiate w'(z) with respect to z such that:

`w''(z) = (4z^3)' => w''(z) = 12z^2`

Hence, evaluating the first order derivative and the second order derivative of function yields  `w'(z) = 4z^3`  and `w''(z) = 12z^2` .

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