Question

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.

Answer 1

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.

Answer 2

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

Answer 3

thank you