anonymous 3 years ago can someone help me simplify the derivative of x/sqrt(1-x^2) ? This is for problem 5A-3G of the second problem sets, and I think my proficiency with radicals may not be up to snuff to be able to properly solve this.

1. anonymous

What do you mean by simplifing? Let's calculate the derivative first:$\frac{ d }{ dx }\frac{ x }{ \sqrt{1-x ^{2}} }=\frac{ d }{ dx }x(1-x ^{2})^{-\frac{ 1 }{ 2 }}=x \frac{ d }{ dx }(1-x ^{2})^{-\frac{ 1 }{ 2 }}+(1-x ^{2})^{-\frac{ 1 }{ 2 }}$$=x(-\frac{ 1 }{ 2 }(1-x ^{2})^{-\frac{ 3 }{ 2 }})\frac{ d }{dx }(1-x ^{2})+(1-x^2)^{-\frac{ 1 }{ 2 }}$$=-\frac{ 1 }{ 2 } x(1-x^2)^{-\frac{ 3 }{ 2 }}(-2x)+(1-x^2)^{-\frac{ 1 }{ 2 }}=\frac{ x^2 }{ \sqrt{(1-x^2)^3} }+\frac{ 1 }{ \sqrt{1-x^2} }$If you want to take that over common denominator, you have to multiply the last term with$\frac{ 1-x^2 }{ 1-x^2}$so you get$\frac{ x^2+1-x^2 }{ \sqrt{(1-x^2)^3} } = \frac{ 1 }{ \sqrt{(1-x^2)^3} }$That might be the simplest form for that derivative.

2. anonymous

thank you very much! that's exactly what I needed!

3. anonymous

thank you