survey of calculus

Question

Find the equation of the tangent line at the given value of x on the curve.

2y3+xy−y=16x4​; x=1

Answer 1

2y³ + xy - y = 16x⁴

When x = 1, 2y³ + y - y = 16. So, 2y³ = 16. Therefore, y = 2.

So, the tangent line intersects the curve at the point (1,2).

Differentiate the given equation implicitly with respect to x:

6y²(dy/dx) + y + x(dy/dx) - dy/dx = 64x³

At the point (1,2), 24(dy/dx) + 2 = 64

dy/dx = 62/24 = 31/12

Equation of tangent line: y - 2 = (31/12)(x - 1)

y = (31/12)x - 7/12