Use implicit differentiation to find the slope of the tangent line to the curve y/(x + 3y) = x^4 + 4 at the point ( 1 , 5/− 14 ) .

Answer 1

(x+3y)y' - y(1+3y')

--------------------- = 4x^3

(x+3y)^2

(x+3y)y' - y(1+3y') = (4x^3)(x+3y)^2

xy' + 3yy' - y - 3yy' = (4x^3)(x+3y)^2

xy' - y = (4x^3)(x+3y)^2

y' = [(4x^3)(x+3y)^2+y] / x

x=1 y = -5/14

y' = [ 4 * ( 9/14)^2 - 5/14]

= 4*81/196 - 70/196

= (324-70)/196

= 254/196

= 127/98