Step by step solution to "What is the equation of the tangent line..."

High School
Calculus and Analysis
Calculus 1
Derivatives

brecruicswbp

Answered question

2022-03-16

What is the equation of the tangent line of

y = {(x - 1)}^{3}

at x=2?

Answer & Explanation

Pentyrch9ci

Beginner2022-03-17Added 9 answers

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 'chain rule'

\frac{d y}{d x} = 3 {(x - 1)}^{2} \cdot \frac{d}{d x} (x - 1) = 3 {(x - 1)}^{2} \cdot 1 = 3 {(x - 1)}^{2}

x = 2 : m = \frac{d y}{d x} = 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

\Rightarrow y = 3 x - 5

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