i need help with this implicit derivative probem?

Question

x^2 + y^2 = 4

ok , so here is my work. well, sort of. I did it in my head since it's a easy one. LOL.
dy^2/dx = -2x

so from here, do we take a square root for both sides?

Answer 1

I am guessing that you want dy/dx, so you want to differentiate everything with respect to x:

Just differentiate every term by x, like so:

d(x^2)/dx + d(y^2)/dx = d(4)/dx:

term 1: We know that the derivative of x^2 = 2x.
term 2: The derivative of y^2 with respect to y is 2y, and by the chain rule, y is a function of x, so we multiply the 2y by (dy/dx).
term 3: The derivative of 4 with respect to x is 0, since 4 is a constant.

Thus we get:

2x + (2y)(dy/dx) = 0

Lastly solve for dy/dx:

2x = -2y(dy/dx)

dy/dx = -x/y

Hope that helped! =)

Answer 2

x^2+y^2=4
take derivative to get
2xdx+2ydy=0
2ydy=-2xdx
ydy=-xdx
ydy/dx=-x
dy/dx=-x/y
y^2=4-x^2, y=sqrt(4-x^2)
dy/dx=-x/sqrt(4-x^2)

Answer 3

x² + y² = 4

Differentiate implicitly.

2x + 2y(dy/dx) = 0
x + y(dy/dx) = 0
y(dy/dx) = -x
dy/dx = -x/y