Posted by Subhotosh Khan on November 12, 2002 at 09:42:28:
In Reply to: Simplifying a solution. posted by Tom Lahay on November 11, 2002 at 14:53:47:
: I have a problem when simplifying one of my answers.
: y'=(1-x^3) * x - 3x^2 * SQRT(1+x^2)
: -----------
: SQRT(1+x^3)
: Simplifying, my book states the first step as:
: x
: y' = ---------- [1 - x^3 - 3x(1+x^2)]
: SQRT(1+x^2)
:
: I see that the result is one less power of 3x and the next expression is no longer square rooted but I am unsure why. I am missing some small algebra trick that I have forgotten over time and my only hypothesis is:
: Because SQRT(1+x^2) = (1+x^2)^1/2
: If I mutiply the power of -3x^2 by the power of the expression (1+x^2)^1/2 then it may follow that:
: -3x(1+x^2) since 2*1/2=1
: Is it because 1/2 is a common factor or what? Any proofs, explanations would be appreciated.
: Then next step makes complete sense to me but I would like the full picture.
: Thanks all.
: Tom L
************************************
you wrote:
y'=(1-x^3)*x/[SQRT(1+x^3)]- 3x^2*SQRT(1+x^2)..(1)
Simplifying, my book states the first step as:
y' = x/[SQRT(1+x^2)]* [1 - x^3 - 3x(1+x^2)]....(2)
Now the problem
In (1) you have [sqrt(1+x^3)] and (1-x^3). Can you check and confirm those terms?
I think the denominator of the first term in (1) should be [sqrt(1+x^2)].