Question

Consider a curve given implicitly by the equation (1+x)y^3 + (x^4)y - 85 = 0.

A. Calculate dy/dx at a general point (x,y).

B. Write the equation of the tangent line to the curve at the point (3,1).

C. At (3,1), y(x) is defined implicitly as a function of x. Let g(x) be the inverse function of y(x). Compute g'(1).

Answer 1

A
(1+x)y^3 + (x^4)y - 85 = 0
y^3 + 3(1+x)y^2y' + 4x^3y + x^4y' = 0
y' = -(y^3+4x^3y)/(3(1+x)+x^4)
= -(y^3+4x^3y)/(x^4+3x+3)

B
y'(3) = -109/93

So, the tangent line is

y-1 = -109/93 (x-3)

C
y(3) = 1, so g(1) = 3

g'(1) = 1/y'(3) = -93/109