Consider the following for the two (02) items that follow: Let x = - and y = ^4 - ^4 . What is ( dy dx )^2 equal to?

Step-by-step solution

Given x = - and y = ^4 - ^4 . Differentiating x with respect to : dx d = + dx d = ( + ) Differentiating y with respect to : dy d = 4 ^3 ( ) - 4 ^3 (- ) dy d = 4 ^4 + 4 ^3 dy d = 4 ( ^4 + ^4 ) Now, dy dx = dy d dx d = 4 ( ^4 + ^4 ) ( + ) = 4( ^4 + ^4 ) + Squaring both sides: ( dy dx )^2 = 16( ^4 + ^4 )^2 ( + )^2 Using the algebraic identity (a+b)^2 = (a-b)^2 + 4ab , we can rewrite the numerator and denominator: ( + )^2 = ( - )^2 + 4 = x^2 + 4 ( ^4 + ^4 )^2 = ( ^4 - ^4 )^2 + 4 ^4 ^4 = y^2 + 4 Substituting these value