SOLUTION: find the derivative of the following function using the limt defintion
((x-1)^2)/(x)
I can know the answer is 2.(x-1)/(x) but I keep getting it wrong when using the defintion. c
Question 945600: find the derivative of the following function using the limt defintion
((x-1)^2)/(x)
I can know the answer is 2.(x-1)/(x) but I keep getting it wrong when using the defintion. could you show me step-by-step how to get that final answer? Found 2 solutions by Alan3354, rothauserc:Answer by Alan3354(69443) (Show Source):
You can put this solution on YOUR website! find the derivative of the following function using the limt defintion
((x-1)^2)/(x)
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= (x^2 - 2x + 1)/x
= x - 2 + (1/x)
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You can probably do the x term --> 1
The -2 --> 0
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That leaves 1/x
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(1/(x+h) - 1/x)/h = (x - (x+h))/(h*(x^2 + hx))
= -h/(h*(x^2 + hx))
= -1/(x^2 + hx)
Lim = -1/x^2
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--> 1 - (1/x^2)
or (x^2 - 1)/x^2
You can put this solution on YOUR website! the limit definition states,
f'(a) = limit as x approaches a of (f(x) - f(a)) / (x-a) = limit as h approaches 0 of (f(a+h) - f(a)) / h
we are given f(x) = ((x-1)^2)/(x), therefore
(f(a+h) - f(h)) / h = ((a+h-1)^2/(a+h) - (a-1)^2/a) / h
= (a(a+h-1)^2 - (a+h)(a-1)^2) / (a(a+h)h)
= (a(a^2+2ah+h^2-2a-2h+1) - ((a+h)(a^2-2a+1)) / (a^2+ah)h)
= (a^3+2ha^2+ah^2-2a^2-2ah+a-a^3+2a^2-a-ha^2+2ah-h) / (a^2+ah)h)
= (ha^2 -h) / (a^2+ah)h)
= h(a^2-1) / (a^2+ah)h)
= (a^2-1) / (a^2+ah)
now the limit as h approaches 0 = (a^2-1) / a^2
= (a^2 -1) / a^2
= 1 - 1/a^2
= 1 - 1/x^2
