Find the equation to the tangent line of

Question

y = (x^3 -1) * (x^2 + 1)^4 at the point where x = -1?

Answer 1

The curve y = (x³ - 1)×(x² + 1)⁴

Substitute x = -1 in the given curve

y = (-1 - 1)(1 + 1)⁴

Simplify

y = -32

The slope of the curve :m = dy / dx at the point (-1, -32)

Apply the derivative to each side with respective x for given curve

dy / dx = (x³ - 1)4(x² + 1)³ (2x) + 3x²(x² + 1)⁴

dy / dx at x = -1

Substitute x = -1 in the dy / dx

dy / dx = (-1 - 1)4(1 + 1)³ (-2) + 3(1 + 1)⁴

Simplify

dy / dx at x = -1: m = 176

The slope m and the point (x1 , y1) then the tangent line equation : y - y1 = m(x - x1)

Substitute m = 176 and x1 = -1 , y1 = -32 in the tangent line

y - (-32) = 176(x - (-1)

y + 32 = 176(x + 1)

y + 32 = 176x + 176

y = 176x + 144

or

176x - y = -144.