"How do you find the equation of the..." was answered by our community

Zackery Ray

Answered question

2022-02-10

How do you find the equation of the tangent line to each of the following curves at the given point:

(x^{2}) y = x + 2

at (2,1)?

Jaden Petersen

Beginner2022-02-11Added 15 answers

First note that the point (2,1) is on the graph of the equation

x^{2} y = x + 2

since

2^{2} \cdot 1 = 2 + 2

.
The equation

x^{2} y = x + 2

implies that

y = x^{- 1} + 2 x^{- 2}

when

x \neq 0

. The derivative is

\frac{d y}{d x} = - x^{- 2} - 4 x^{- 3}

so that the slope of the tangent line at

(x, y) = (2, 1) i s - 2^{- 2} - 4 \cdot 2^{- 3} = - \frac{1}{4} - \frac{1}{2} = - \frac{3}{4} .

The equation of the tangent line is then

y = f (2) + f^{'} (2) (x - 2) = 1 - \frac{3}{4} \cdot (x - 2),

y = - \frac{3}{4} x + \frac{5}{2} .

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