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ticbol
Posts:
116
Registered:
1/25/05
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Re: Chain Rule, Simplify General Power Rule
Posted:
Apr 23, 2005 2:43 AM
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Actually, there is a skip berween your step 3 and step 4.
Let us call it "step 3-4"
Or, for that matter, there are many skips, if I can call them skips, in
the solution.
The solution, as shown, is done by someone who is good enough to do
"mental" simplifications, or simplifications not shown "on paper".
------------
Let me supply these skips.
>I have. f(x) = x^2 (sqrt(1 -x^2))
>step 1: rewrite: f(x) = x^2 (1 -x^2) ^1/2
>step 2: Prod Rule: f'(x) = [x^2]*[d/dx[(1 -x^2)^1/2] + [(1
-x^2)^1/2]*[d/dx(x^2)]
I am using parenthesers(..) and brackets[..] as separators and/or
multiplicators freely.
>step 3: General Power Rule: = [x^2]*[(1/2) ((1 -x^2)^ -1/2) (-2x)] +
[(1 -x^2)^1/2]*[2x]
Simplify:
= [(x^2)(1/2)(-2x)][(1 -x^2)^ -1/2] +(2x)[(1 -x^2)^ 1/2]
= (-x^3)[(1 -x^2)^ -1/2] +(2x)[(1 -x^2)^ 1/2]
>step 4: Simplify: = - x^3 [(1 -x^2)^ -1/2] + 2x (1 - x^2) ^-1/2
Whoa.
How did you get the second term/expression [2x (1 -x^2)^ -1/2] ?
Why, all of a sudden, the former surd [(1 -x^2)^ 1/2] became [(1 -x^2)^
-1/2] ?
Or, why, all of a sudden, sqrt(1 -x^2) became 1/[sqrt(1 -x^2)] ?
That is not correct.
If that "step 4" is in the book, then the guy who did the solution made
a big leap/skip here.
------------
I see that the final answer shown in "step 6" below is the form of
sqrt(1 -x^2). Then, why don't we use that from here on. It would be
easier to "visualize" our solution.
To continue,....
= (-x^3)[(1 -x^2)^ -1/2] +(2x)[(1 -x^2)^ 1/2]
= (-x^3)/[sqrt(1 -x^2)] +2x[sqrt(1 -x^2)]
Combine the two terms into one, using sqrt(1 -x^2) as the common
denominator,
= [-x^3 +2x[sqrt(1 -x^2)]*sqrt(1 -x^2)] / [sqrt(1 -x^2)]
= [-x^3 +2x[sqrt(1 -x^2)]^2] / [sqrt(1 -x^2)]
= [-x^3 +2x(1 -x^2)] / [sqrt(1 -x^2)]
= [-x^3 +2x -2x^3] / [sqrt(1 -x^2)]
= [-3x^3 +2x] / [sqrt(1 -x^2)]
We can stop there.
But if you want to simplify further, to conform with the answer shown
in "step 6" below,
= [x(-3x^2 +2)] / [sqrt(1 -x^2)]
= [x(2 -3x^2)] / [sqrt(1 -x^2)]
There.
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>step 5: Factor least power: = x(1 - x^2) ^ -1/2 [- x^2 (1) + 2 (1 -
x^2)]
>step 6: Simplify: = (x (2 - 3x^2)) / (sqrt(1 - x^2))
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