Question 1193267

Given the equation {{{x^2+y^2=9}}} , express {{{y}}} explicitly and calculate the
derivative.

 {{{x^2+y^2=9}}} 

 {{{y^2=9-x^2}}} 

 {{{y=sqrt(9-x^2)}}} 


calculate the derivative:


 {{{(d/dx)(sqrt(9-x^2))}}}


 {{{(d/dx)((9-x^2)^(1/2))}}}............apply the chain rule


={{{(1/(2sqrt(9-x^2)))(d/dx)(9-x^2)}}}


={{{(1/(2sqrt(9-x^2)))(-2x)}}}


={{{(1/(sqrt(9-x^2)))(-x)}}}.....factor {{{9-x^2}}}


= {{{-x/sqrt((3 - x)(3+x))}}}


= {{{-x/(sqrt(3 - x) sqrt(x + 3))}}}

so,

{{{y}}}'{{{(x) = -x/(sqrt(3 - x) sqrt(x + 3))}}}