Implicit Differentiation

Question

Find the slope of the graph of the relation x²y + 4y = 8 at the point (2, 1).

Options:

1. 1
2. 3/2
3. 1/2
4. -1/2

Answer 1

Start by taking d/dx of both sides

x²y' + y·(2x) + 4y' = 0

x²y' + 2xy + 4y' = 0

x²y' + 4y' = -2xy

y' (x² + 4) = -2xy

y' = -2xy/(x² + 4)

Now evaluate y'(2,1)

y'(2,1) = [-2(2)(1)]/(2² + 4) = -4/8 = -1/2

Answer 2

x²y + 4y = 8

2x*y + x²*y' + 4y' = 0

(x² + 4)*y' = -2x*y

y' = -2x*y / (x² + 4)

At (2,1):

y' = -2(2)(1) / 8 = -1/2