Question

What is the equation of the tangent line of `y=(x-1)^3` at `x=2`?

Answer 1

y = 3x - 5

Explanation:

The equation of the tangent is of the form: y - b = m ( x - a )

where m represents the gradient (slope) of the line and

(a , b ) a point on the line . The x-coordinate is known and

to find y substitute x = 2 into the equation. To find m

differentiate the function and evaluate it for x = 2.

Using `color(blue)(" ' chain rule '")`

`dy/dx = 3(x - 1 )^2 .d/dx (x - 1 ) = 3(x - 1 )^2 .1 = 3(x - 1 )^2`

x = 2 : `m = dy/dx = 3( 2 - 1 )^2 = 3`

and y = `(2 - 1 )^3 = 1`

equation of tangent: y - b = m (x - a ) , m = 3 , (a , b )= (2 , 1 )

hence y - 1 = 3 (x - 2 ) so y - 1 = 3x - 6

`rArr y = 3x - 5`