Question

Problem 5A-3g, is the answer correct?

Answer 1

I'm not able to get the same nice clean answer as in the solution, and I think the problem is a mistake in the given solution where he get

\[\frac{dy}{dx} \frac{x}{\sqrt{1 - x^2}} = (1-x^2)^{-3/2} ??\]
it seems like he forgot the \(x\) in the numerator
I get
\[=\frac{-x^2+x+1}{(1-x^2)\sqrt{1-x^2}}\]

What am I missing, where is my mistake ?

Answer 2

First change the equation to \[\frac{ dy }{ dx}x(x-x ^{2})^{-\frac{ 1 }{ 2 }}\]
Use the product rule you get \[(1)(1-x ^{2})^{-\frac{ 1 }{ 2 }}+x(-\frac{ 1 }{ 2 })(1-x ^{2})^{-\frac{3 }{ 2 }}(-2x)\]
this can be simplified to \[\frac{ 1-x ^{2} }{(1-x ^{2})^{\frac{3}{2}} }+\frac{ x ^{2} }{(1-x ^{2})^{\frac{3}{2}} }\]
which equals \[\frac{ 1 }{ (1-x ^{2} )^{\frac{ 3 }{ 2 }}}\] or\[(1-x ^{2})^{\frac{ -3 }{ 2 }}\]

Answer 3

Note that \[(1-x ^{2})^{-\frac{ 1 }{ 2 }}=\frac{ 1 }{ (1-x ^{2})^{\frac{ 1 }{ 2 }} }=\frac{ 1-x ^{2} }{ (1-x ^{2})^{ \frac{ 3 }{ 2 } }}\]

Answer 4

oh man.... you are right :). I have to go back and check my calculation, but I think my silly mistake was to multiply by (-2), not (-2x), arghhh... sometime you just go blind staring at a problem.