If y+x + y-x = c such that dy dx = f(x) - [f(x)]^2 - 1 , then f(x) = ____

Step-by-step solution

Given equation: y+x + y-x = c Differentiating both sides with respect to x : 1 2 y+x ( dy dx + 1 ) + 1 2 y-x ( dy dx - 1 ) = 0 dy dx ( 1 y+x + 1 y-x ) = 1 y-x - 1 y+x dy dx ( y-x + y+x y^2-x^2 ) = y+x - y-x y^2-x^2 dy dx = y+x - y-x y+x + y-x Rationalizing the denominator by multiplying the numerator and denominator by y+x - y-x : dy dx = ( y+x - y-x )^2 ( y+x )^2 - ( y-x )^2 dy dx = (y+x) + (y-x) - 2 y^2-x^2 (y+x) - (y-x) dy dx = 2y - 2 y^2-x^2 2x dy dx = y x - y^2-x^2 x dy dx = y x - ( y x )^2 - 1 Comparing this