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PLEASE HELP Will award medal and fan for an explanation to this question! Question in comments (Involves simplifying (f(x-h)-f(x))/h

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Given \[f(x)=-3x+\frac{ 1 }{ x }-3 \] Simplify \[\frac{ f(x-h)-f(x) }{ h }\] When x=4
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I just need to simplify (f(x-h)-f(x))/h when x=4

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another way to write \[-3x+\frac{ 1 }{ x }-3\] is \[\frac{ -3x ^{2}-3x+1 }{ x }\] so then f(x+h) is \[\frac{ -3(x+h)^{2}-3(x+h)+1 }{ x+h }\] and then f(x+h)-f(x) is \[\frac{ -3(x ^{2}+2xh+h ^{2})-3x-3h+1 }{ x+h }-\frac{ -3x ^{2}-3x+1 }{ x }\] and then I give them the same denominator so that I can combine them\[\frac{ -3x ^{3}-6x ^{2}h-3xh ^{2}-3x ^{2}-3xh+x }{ x(x+h) }+\frac{ (x+h)(3x ^{2}+3x-1) }{ x(x+h) }\]and then multiply the numerator of the second part\[\frac{ 3x ^{3}+3x ^{2}h+3x ^{2}+3xh-x-h }{ x(x+h) }\] and then cancel out as much as i can\[\frac{ -3x ^{2}h-3xh ^{2}-h }{ x(x+h) }\]and then I take an h out of the numerator\[\frac{ h(3x ^{2}-3xh-1) }{ x(x+h) }\] so basically my question is: how to I simplify:\[\frac{ \frac{ h(3x ^{2}-3xh-1) }{ x(x+h) } }{ h }\]
To simplify, start by just substituting 4 in for every x. Then follow your order of operations.
@marihelenh quick question, how to I get rid of the bottom h? would I multiply it by the denominator in the numerator? in other words, would I multiply x(x+h) by h to get a simple fraction instead of a fraction inside of a fraction?
If you recall, when you divide fractions, you can multiply by the reciprocal and it will give you the same thing. So, you could multiply the top part by 1/h. |dw:1437424044483:dw|
have you thought of doing this term by term ?! that way, the 3 drops out immediately and you therefore get closer to the answer that much more quickly. what you are trying to do here is calculate the derivative of your function \(f(x)\). and constants "drop out" :p so work out f(x-h) then work out f(x-h) - f (x) only then divide by h
answer choices are: a) \[\frac{ 49-12h }{ 16-4h }\] b) \[0\] c) \[\frac{ -3h2-3h+1 }{ h }\] d) \[\frac{ -59+4h }{ 4 }\] e) \[\frac{ -49-12h }{ 16+4h }\]
@IrishBoy123 could you please explain to me how to do that? Someone else taught me how to do it that way but a faster way would be much appreciated!!
i cannot do it for you but: \( \large f(x)=−3x+ \frac{1}{x}−3\) so, \( \large f(x-h)=−3(x-h)+ \frac{1}{x-h}−3\) and so you can see that working out \( \large f(x-h)- f(x)\) should already be a lot easier once you do that, you can think about dividing 'h' in

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