Use implicit differentiation to find an equation of the tangent line to the curve at the given point. 4x2 + xy + 4y2 = 9, (1, 1) (ellipse)

Answer 1

4x² + 4y² +xy = 9 @ (1,1)

taking the derivative of with respect to x in y gives us dy/dx, so let dy/dx = y'

4x² (d/dx) +4y² (d/dx)+xy (d/dx) = 9 (d/dx)

8x +8y*y'+(y+xy') = 0

rearrange:

y'(8y+x) = 8x-y

y' = (8x-y)/(8y+x) <<< Plug in values to find the slope at (1,1)

y' = [8(1)-(1)]/[8(1)+(1)]

y' = 7/9 at (1,1). This is our m, or slope. Now we find our intercepts using the slope-intercept form:

y-y₁ = m(x-x₁)

y-1=7/9 (x-1)

y= 7/9 (x-1) + 1

y= 7/9(x) +2/9 << Final Answer>>