Question

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Answer 1

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Answer 2

First, find the slope at x=0
Do you know how to use the quotient rule?

Answer 3

honestly im fairly lost on the topic if you could show how or walk me through it thatd be great

Answer 4

OK. The quotient rule says that, given f(x) and g(x):
\[\frac{d}{dx} \frac{f(x)}{g(x)} = \frac{f'(x)g(x)-g'(x)f(x)}{g(x)^2}\]

Can you find f(x) and g(x) in the given equation:
\[\frac{x^2-4}{x^2+4}\]

Answer 5

g(x) x^2

Answer 6

f(x) same thing?

Answer 7

Not quite.
F(x) will be the top part of the fraction. So, f(x )= x²-4, and g(x) = x²+4

Answer 8

Knowing f(x) and g(x), we can find g²(x), f'(x) and g'(x):
\[f'(x) = 2x \\ g'(x) = 2x \\ g(x)^2=(x^2+4)^2\]

Answer 9

Then, we plug all of those values into the quotient rule to get:
\[\frac{ 2x(x^2+4)-2x(x^2-4) }{ (x^2+4)^2 } = \frac{ 2x((x^2+4)-(x^2-4)) }{ x^4+8x^2+16 } = \frac{ 16x }{ x^4+8x^2+16 }\]

Answer 10

that's not the equation though?

Answer 11

That's just the derivative of the function.

Answer 12

To find the slope at (0,-1), evaluate that at x=0. Once you have the slope, just plug it into the point slope equation:
y-y₁=m(x-x₁)

Answer 13

ok you lost me there, if you plug 0 in for x for the derivative you get a 0 numerator

Answer 14

So then the slope of the function at x=0 is 0. (A flat line)

Answer 15

ok, so that makes it y=-1? to get the y coordinate?

Answer 16

the equation is y=-1 i mean

Answer 17

Yup

Answer 18

ty for making sense of this appreaciate it