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Topic: Chain Rule, Simplify General Power Rule
Replies: 2   Last Post: Apr 23, 2005 5:04 AM

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Posts: 116
Registered: 1/25/05
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

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]

= [(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


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
= [-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)]


>step 5: Factor least power: = x(1 - x^2) ^ -1/2 [- x^2 (1) + 2 (1 -

>step 6: Simplify: = (x (2 - 3x^2)) / (sqrt(1 - x^2))

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