Differentiate both sides of
y³ + xy² - 4 = x + 4y²
with respect to x.
Using the power, product, and chain rules,
d/dx [y³] = 3y² dy/dx
d/dx [xy²] = d/dx [x] y² + x d/dx [y²] = y² + 2xy dy/dx
d/dx [4y²] = 8y dy/dx
Putting everything together, we have
3y² dy/dx + y² + 2xy dy/dx = 1 + 8y dy/dx
Solve for dy/dx :
(3y² + 2xy - 8y) dy/dx = 1 - y²
dy/dx = (1 - y²) / (3y² + 2xy - 8y)
From the given equation, when x = 4, we have
y³ + 4y² - 4 = 4 + 4y²
y³ = 8
y = 2
so that the curve passes through the point (4, 2).
When x = 4 and y = 2, the derivative has a value of
dy/dx (4, 2) = (1 - 2²) / (3•2² + 2•4•2 - 8•2) = -1/4
and this is the slope of the tangent line.
Use the point-slop formula to get the line's equation:
y - 2 = -1/4 (x - 4)
y - 2 = -1/4 x + 1
y = -1/4 x + 3