Evaluate the derivative of the function at the given point.? Use a graphing utility to verify your result. (If an answer is undefined, enter UNDEFINED. Round your answer to three decimal places.)`f(x)=3/(x^2-4x)^2`  at `(5,3/25)`  

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Chain rule:


Sum rule:

`(f(x)pm g(x))'=f'(x)pm g'(x)`

So by using the above rules we can differentiate our function, but first let's rewrite function `f` to get more convenient form.



So to calculate `f'(5)` you simply put 5 instead of `x.`


I have to point out that this is not rounded result, that is there are only 3 decimal places, no more.

Now to be able to graph the tangent at point `(5,3/25)`  we must find equation of the tangent line which is given by formula:

`y=f'(x)x+l`                                                                       (1)

Now to calculate `l` we simpliput all data we have into formula (1).




Blue line represents your function `f` while red line represents tangent at point `(5,3/25)`.

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To evaluate the derivative, it is best to rearrange the function slightly to make it easier to take the derivative.



Now using the power rule and chain rule:

`f'(x)=3(-2)(x^2-4x)^{-3}(2x-4)`  and evaluate at x=5



The derivative at x=5 is `f'(5)=-36/125` .

The graph and its derivative is:

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