how to find the dy/dx by implicit differentiation

Question

5x²+y²=4

dy/dx=

Answer 1

Breaking this into 3 pieces:

1) The derivative of 5x² with respect to "x" is, using the power rule, 10x

2) The derivative of y² with respect to "x" is, using the power rule, 2y, however, we must use the chain rule, and since "y" is some complex function of "x" that we do not know, we can say the derivative of what is "inside" is dy/dx making the result: 2y•dy/dx

3) The derivative of the constant "4" with respect to "x" is zero (the derivative of all constants is zero)

So the result is: 10x + 2y•dy/dx = 0 then subtracting 10x from both sides we get:

2y•dy/dx = -10x then dividing both sides by "2y" we get:

dy/dx = -10x/(2y) = -5x/y

Answer 2

dx(5x²+y²) = dx(4)

dx(5x²) + dx(y²) = 0

5 * 2x + 2y * dx(y) = 0

10x + 2y * y' = 0

2y * y' = -10x

then divide by 2y

y' = -10x/(2y)

y' = -(5x)/y