`f(x) =1/(x^2 - 3x)^2, (4,(1/16))` Find and evaluate the derivative of the function at the given point. Use a graphing utility to verify your result.

Textbook Question

Chapter 2, 2.4 - Problem 68 - Calculus of a Single Variable (10th Edition, Ron Larson).
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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the derivative of the function, using the chain and quotient rules, such that:

`y' = (-2(x^2 - 3x)(x^2 - 3x)')/((x^2 - 3x)^4) => y' = (-2(x^2 - 3x)(2x - 3))/((x^2 - 3x)^4)`

Simplifying by `(x^2 - 3x)` yields:

`y' = (-2(2x - 3))/((x^2 - 3x)^3)`

Now, you need to evaluate the value of derivative at x = 4:

`y' = (-2(2*4 - 3))/((4^2 - 3*4)^3)`

`y' = -10/64`

`y' = -5/32`

Hence, evaluating the derivative of the function yields `y' = (-2(2x - 3))/((x^2 - 3x)^3)` and evaluating the value of derivative at x = 4, yields `y' =- 5/32` .

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