What is the derivative of this function?Please show how it can be done via power rule, and via product rule(2x-1)^2(X^2-9)

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embizze | High School Teacher | (Level 1) Educator Emeritus

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(1) Using only the product rule, find `d/(dx)[(2x-1)^2(x^2-9)]`

(a)Rewrite as `d/(dx)[(2x-1)(2x-1)(x^2-9)]`

Use the extended product rule `d/(dx)f*g*h=f'gh+fg'h+fgh'` where f,g, and h are differentiable functions of x.

Thus `d/(dx)[(2x-1)(2x-1)(x^2-9)]`

`=2(2x-1)(x^2-9)+(2x-1)(2)(x^2-9)+(2x-1)(2x-1)(2x)`

`=4x^3-2x^2-36x+18+4x^3-2x^2-36x+18+8x^3-8x^2+2x`

`=16x^3-12x^2-70x+36`

(b) Or rewrite as `d/(dx)[(2x-1)(2x-1)(x+3)(x-3)]` , but this is just more work. You would use `d/(dx)f*g*h*k=f'ghk+fg'hk+fgh'k+fghk'`

(2) Using the power rule `d/(dx)u^n=n(u^(n-1))(du)/(dx)` (Note the use of the chain rule):** we still use the product rule**

`d/(dx)[(2x-1)^2(x^2-9)]`

`=2(2x-1)(2)(x^2-9)+(2x-1)^2(2x)`

`=(8x-4)(x^2-9)+(4x^2-4x+1)(2x)`

`=8x^3-4x^2-72x+36+8x^3-8x^2+2x`

`=16x^3-12x^2-70x+36`

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Either way the derivative is `16x^3-12x^2-70x+36`

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