Guse implicit differentiation to find an equation of the tangent line to the curve at the given point. x2/3 + y2/3 = 4 (−3 3 , 1) (asteroid)

X²/3 + y²/3 = 4
start by multiplying everything by 3 to get rid of fractions

x² + y² = 12

(it's a circle)
take derivative with respect to x

2x + 2yy' = 0

solve for y' now

2x = -2yy'

2x/(-2y) = y'

-x/y = y'

use the given point (-33,1)

-(-33)/1 = y'

y' = 33

y' is rate of change or slope at that point. Now you can use the point (-33,1) again to find y-intercept "b" for the tangent line.

Line so far:
y = 33x + b

plug in (-33,1)
1 = 33(-33) + b

1 = -1089 + b
1090 = b

tangent line is:
y = 33x + 1090

Iwould say always for the both of them