Question

Find the equation of the tangent line. y=(9-x^2)^1/3 at (1,2)​

Answer 1

Get the derivative:

y = (9 - x²)¹ʹ³

dy/dx = 1/3 (9 - x²)⁻²ʹ³ d/dx [9 - x²]

dy/dx = 1/3 (9 - x²)⁻²ʹ³ (-2x)

dy/dx = -2/3 x (9 - x²)⁻²ʹ³

Evaluate it at x = 1 :

dy/dx (1) = -2/3 • 8⁻²ʹ³

Since 8 = 2³, we have

8⁻²ʹ³ = 1 / 8²ʹ³ = 1 / (2³)²ʹ³ = 1 / 2² = 1/4

Then the tangent line has equation

y - 2 = 1/4 (x - 1) → y = 1/4 x + 7/4