anonymous
  • anonymous
What is the instantaneous slope of y = (-2/x) at x = 2?
Mathematics
  • Stacey Warren - Expert brainly.com
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katieb
  • katieb
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
anonymous
  • anonymous
-1 -1/2 1 1/2
DanJS
  • DanJS
d/dx
mathmale
  • mathmale
The derivative gives you an expression whose meanings include "inst. slope of the tangent line."

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mathmale
  • mathmale
Given y = (-2/x), find the derivative, dy/dx.
anonymous
  • anonymous
i think that its 1/2?
mathmale
  • mathmale
\[\frac{ dy }{ dx }=\frac{ d }{ dx }(-\frac{ 2 }{ x })\]
anonymous
  • anonymous
im using the old derivative formula
mathmale
  • mathmale
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.
anonymous
  • anonymous
substitute 2 for x and you will get \[y=-2/(2)\]
anonymous
  • anonymous
( f(a+h) - f(a) ) /h
mathmale
  • mathmale
@hippo211 Your statement is true. However, our goal is to find the derivative of y, not the value of the function.
mathmale
  • mathmale
I see. You're using the "difference quotient" definition of the derivative.
DanJS
  • DanJS
limit for h goes to 0
mathmale
  • mathmale
Have you studied derivative formulas such as \[\frac{ d }{ dx }x^n=nx ^{(n-1)}?\]
anonymous
  • anonymous
(-2/(2+h)) + 1 / h ( h/(2+h) ) * 1/h 1/2+h 1/2?
anonymous
  • anonymous
no
mathmale
  • mathmale
So, you have the function f(x)=-2/x, or \[f(x)=\frac{ -2 }{ x }\]
mathmale
  • mathmale
What is f(x+h)?
mathmale
  • mathmale
Substitute (x+h) into the function f(x)=-2/x.
anonymous
  • anonymous
-2 / (2+h)
mathmale
  • mathmale
In other words, replace "x" with "x+h"
mathmale
  • mathmale
Then you want to find the limit as h goes to zero of the following:
mathmale
  • mathmale
\[\frac{ f(x+h)-f(x) }{ h }\]
mathmale
  • mathmale
Here you have the correct f(x+h). It is \[f(x+h)=\frac{ -2 }{ x+h }\]
mathmale
  • mathmale
To find the derivative of the function f(x)=-2/x, evaluate the limit (as h goes to zero) of
mathmale
  • mathmale
\[\frac{ \frac{ -2 }{ x+h }-[\frac{ -2 }{ x }]\ }{ h }\]
mathmale
  • mathmale
Can you do that?
mathmale
  • mathmale
Hint: the two fractions in the numerator have different denominators, so you must find and use the LCD to combine them into one fraction.
anonymous
  • anonymous
that is what i did ^ but you said it was wrong
mathmale
  • mathmale
I don't recall having used the word "wrong" in our discussion.
anonymous
  • anonymous
so did i do it correct?
mathmale
  • mathmale
If the 2 different denominators are x and (x+h), then what is the LCD?
anonymous
  • anonymous
(x+h)
mathmale
  • mathmale
Actually, you must multiply x by (x+h) to obtain the LCD. Do that now, please.
mathmale
  • mathmale
Don't multiply that out; simply write x(x+h).
mathmale
  • mathmale
We now have to take the limit of the following as h approaches zero:
mathmale
  • mathmale
\[\frac{ f(x+h)-f(x) }{ h }\]
mathmale
  • mathmale
And since f(x)=-2/x, that comes out as follows: Find the limit of the following as h goes to zero:
mathmale
  • mathmale
\[\frac{ \frac{ -2 }{ x+h }-\frac{ -2 }{ x } }{ h }\]
mathmale
  • mathmale
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:
mathmale
  • mathmale
\[\frac{ \frac{ x(-2) }{ h }-\frac{ x+h }{ ? } }{ ? }\]
mathmale
  • mathmale
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?
mathmale
  • mathmale
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.
mathmale
  • mathmale
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.

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