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Find the change in y and the differential dy.

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I see we're really working on Calculus today, Dalia! :)
 
Alright, let's tackle this one. First, a rewrite...
 
y = 4 x1/2  (Derivation doesn't like roots; it likes exponents.)
 
Now the derivative...
 
dy = 2 x-1/2 dx
 
Solving the differential at the point given...
 
dy = 2 (2)-1/2(0.4)
    = (2*0.4)/√2
    = (0.8)/√2 OR 4 / (5√2) OR .56569.

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thank you so much for your help! to find the change in y for the first part of the question, do i simply plug in the given numbers into the derivative like youve done?
I honestly find it a bit odd that they ask you that before the second question, but it depends on the interpretation of the first. 
 
At x=2, we know y=4 sqrt 2.
We found at that point that y iischanging at a rate of 0.56569. So at that point, that is the change in y.
 
With an alternate interpretation, you can take two points near x=2 and just compute both y values and subtract them. For instance, find the y values for x=1.9 and x=2.1 using the original equation. Subtract those two values to find the change in y.

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