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

Charlotte Holden

Answered question

2022-04-15

How do you find the equation of the tangent line to the curve

y = \frac{| x |}{\sqrt{2 - x^{2}}}

at (1,1)?

Varikoseocaz

Beginner2022-04-16Added 9 answers

For x near 1, x, 0, so |x|=x and

y = \frac{x}{\sqrt{2 - x^{2}}}

Using the quotient rule for derivatives and the chain rule to differentiate

\sqrt{2 - x^{2}}

,
we get

y^{'} = \frac{(1) (\sqrt{2 - x^{2}}) - (x) (\frac{1}{2 \sqrt{2 - x^{2}}} \cdot - 2 x)}{{(\sqrt{2 - x^{2}})}^{2}}

= \frac{\sqrt{2 - x^{2}} + \frac{x^{2}}{\sqrt{2 - x^{2}}}}{2 - x^{2}}

We could rewrite some more, but we're really only interested in y' at x=1, so let's just do the arithmetic.
At x=1, the slope of the tangent line is

m = \frac{\sqrt{2 - {(1)}^{2}} + \frac{{(1)}^{2}}{\sqrt{2 - {(1)}^{2}}}}{2 - {(1)}^{2}} = \frac{1 + 1}{1} = 2

The equation of the line through (1,1) with slope 2 is
y-1=2(x-1) or
y=2x-1

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