Question

What is the instantaneous slope of y = (-2/x) at x = 2?

Answer 1

-1
-1/2
1
1/2

Answer 2

d/dx

Answer 3

The derivative gives you an expression whose meanings include "inst. slope of the tangent line."

Answer 4

Given y = (-2/x), find the derivative, dy/dx.

Answer 5

i think that its 1/2?

Answer 6

\[\frac{ dy }{ dx }=\frac{ d }{ dx }(-\frac{ 2 }{ x })\]

Answer 7

im using the old derivative formula

Answer 8

Sorry, it's not 2. The derivative of 2x is 2. The derivative of 1/(2x) is very different. Show me your "old derivative formula," please.

Answer 9

substitute 2 for x and you will get \[y=-2/(2)\]

Answer 10

( f(a+h) - f(a) ) /h

Answer 11

@hippo211 Your statement is true. However, our goal is to find the derivative of y, not the value of the function.

Answer 12

I see. You're using the "difference quotient" definition of the derivative.

Answer 13

limit for h goes to 0

Answer 14

Have you studied derivative formulas such as
\[\frac{ d }{ dx }x^n=nx ^{(n-1)}?\]

Answer 15

(-2/(2+h)) + 1 / h
( h/(2+h) ) * 1/h
1/2+h
1/2?

Answer 16

no

Answer 17

So, you have the function f(x)=-2/x, or \[f(x)=\frac{ -2 }{ x }\]

Answer 18

What is f(x+h)?

Answer 19

Substitute (x+h) into the function f(x)=-2/x.

Answer 20

-2 / (2+h)

Answer 21

In other words, replace "x" with "x+h"

Answer 22

Then you want to find the limit as h goes to zero of the following:

Answer 23

\[\frac{ f(x+h)-f(x) }{ h }\]

Answer 24

Here you have the correct f(x+h). It is \[f(x+h)=\frac{ -2 }{ x+h }\]

Answer 25

To find the derivative of the function f(x)=-2/x, evaluate the limit (as h goes to zero) of

Answer 26

\[\frac{ \frac{ -2 }{ x+h }-[\frac{ -2 }{ x }]\ }{ h }\]

Answer 27

Can you do that?

Answer 28

Hint: the two fractions in the numerator have different denominators, so you must find and use the LCD to combine them into one fraction.

Answer 29

that is what i did ^ but you said it was wrong

Answer 30

I don't recall having used the word "wrong" in our discussion.

Answer 31

so did i do it correct?

Answer 32

If the 2 different denominators are x and (x+h), then what is the LCD?

Answer 33

(x+h)

Answer 34

Actually, you must multiply x by (x+h) to obtain the LCD. Do that now, please.

Answer 35

Don't multiply that out; simply write x(x+h).

Answer 36

We now have to take the limit of the following as h approaches zero:

Answer 37

\[\frac{ f(x+h)-f(x) }{ h }\]

Answer 38

And since f(x)=-2/x, that comes out as follows: Find the limit of the following as h goes to zero:

Answer 39

\[\frac{ \frac{ -2 }{ x+h }-\frac{ -2 }{ x } }{ h }\]

Answer 40

Recall that the LCD is x(x+h). To get that in the first fraction, multiply -2/(x+h) by x and divide the whole thing by x:

Answer 41

\[\frac{ \frac{ x(-2) }{ h }-\frac{ x+h }{ ? } }{ ? }\]

Answer 42

Sorry, that's not complete. Takes a while to type out something like this in Equation Editor. Are you able to complete this work yourself?

Answer 43

I know this is intense and long, but I need y our attention. OpenStudy says you're "just looking around." Want to finish this problem or not? I need your active participation.

Answer 44

Let me know when you're ready to continue, and then either someone else or I will help you complete the operation of finding the derivative of -2/x.