Determine the equation of the tangent to the curve 3y^(2)x − 2x^3 = 5xy at the point P(1,2).

Question

just want to clarify 3y^(2)x is 3y^2 times x

Answer 1

Implicit differentiation (remember product rule and any time we differentiate a y term we put a y' after):

3xy² - 2x³ = 5xy

3(y² + 2xyy') - 6x² = 5(y + xy') at (1,2)

3(4 + 4y') - 6 = 5(2 + y')

12 + 12y' - 6 = 10 + 5y'

7y' = 4

y' = 4/7