Question
Implicit Differentiation: Find dy/dx for each. Leave answers in terms of both x and y.
1) 1/y + 1/x = 1 + y^2
2) -(x^2 / y^2)
1) 1/y + 1/x = 1 + y^2
2) -(x^2 / y^2)
Answers
Steve
1/y + 1/x = 1 + y^2
-1/y^2 y' - 1/x^2 = 2y y'
(-1/y^2 - 2y)y' = 1/x^2
y' = (1/x^2)/(-1/y^2 - 2y)
= -(1/x^2)/((1+2y^3)/y^2)
= -y^2 / (x^2(1+2y^3))
-(x^2/y^2)
-x^2 y^-2
-2x y^-2 + 2x^2y^-3 y'
2x(x-y)/y^3 y'
no equation, but that's where we are so far.
-1/y^2 y' - 1/x^2 = 2y y'
(-1/y^2 - 2y)y' = 1/x^2
y' = (1/x^2)/(-1/y^2 - 2y)
= -(1/x^2)/((1+2y^3)/y^2)
= -y^2 / (x^2(1+2y^3))
-(x^2/y^2)
-x^2 y^-2
-2x y^-2 + 2x^2y^-3 y'
2x(x-y)/y^3 y'
no equation, but that's where we are so far.