Question

Can someone please walk me through this??
Find the derivative of f(x) = -9/x at x = 6

Answer 1

sure

Answer 2

what is the y value of that function when x = 6?

Answer 3

the -9 is a constant, so :\[\huge f(x)=\frac{-9}{x}\]\[\huge f'(x)=-9*\frac{d}{dx}\frac{1}{x}\]

Answer 4

the y value is -3/2 right?

Answer 5

This is the appropriate approach to finding the derivative of the given function:\[f '(x)=\frac{ -9 }{ 1 }\frac{ d }{ dx }\frac{ 1 }{ x }\] with acknowledgement and thanks to cleats.
Hint: rewrite the function 1/x before attempting to find the derivative.

Answer 6

Keep in mind that we must find the derivative before substituting x=6.

Answer 7

You could also do the quotient rule but that would be silly

Answer 8

But what do I substitute into the formula??

Answer 9

Until you have the derivative, nothing.

Hint: what is the derivative of (1/x)?
Hint: rewrite (1/x) as x^(-1):\[\frac{ 1 }{ x }=x ^{-1}\]

Answer 10

\[\frac{ d }{ dx }x ^{-1}=?\]

Answer 11

Hint: Power Rule for differentiation.

Answer 12

i don't think she need to differentiate, she just need to differentiate at a point.

Answer 13

to do that you simply find the value of y at that point. Then when you have done that find the value of y at a point close to it (for example at x=6.0001) and then compare the value of y.

If you then take the difference of the second y and the first y and divide by the difference in x (in my example 0.0001) then you'll have the approximate slope at that point.

Answer 14

I'm so confused.. I don't know what to do with the formula at all.

Answer 15

Thanks for the input, but the problem does state, "Find the derivative .... " and then adds, "at x=6)."

We must find the derivative first, and then substitute x=6, to answer this question.

Were we to substitute x=6 right away, the result would be a constant, whose derivative would be zero.

Answer 16

@nikin, I'm really sorry for the confusion.

What do you know about the power rule for differentiation?

Can you differentiate y=x^2?

Answer 17

no mathmale to find the derivative at a point you don't need to take the derivative, she probably hasn't even touched the subject of differentials yet.

Answer 18

I don't even know what differentiate means

Answer 19

I think she hasn't seen a table of derivatives yet either

Answer 20

@mathmale \[\frac{ \Delta y }{ \Delta x }=\frac{ y_2-y_1 }{ 6.0001-6 }\]

Answer 21

"differentiate" means "find the derivative of".

What course are you in, @nikin?

Answer 22

pre calculus with flvs

Answer 23

yes mathmale we both know that but if you want to teach her how to differentiate when she likely never even heard about that, you'll only confuse her

Answer 24

she does not need to differentiate the formula only take an approximation of the point

Answer 25

He's talking about the wording of the question confusing us rather than teaching her derivatives

Answer 26

These are the answer choices
2/3
1/4
3/2
4

Answer 27

In that case, the others are correct in suggesting a different approach. The question then becomes WHICH method? I have to take back my suggestion "differentiate" and am sorry it confused you.

@zimmah, how about demonstrating how to find the slope of the SECANT line connecting the given point, (6, f(6)), with another point on the graph, such as (6.1, f(6.1))?

Answer 28

Note that the problem statement DOES ask for "the derivative." I was remiss only in that I did not ask about the content in which you found this problem. Still, you must understand that "derivative" in this case pertains to the slope of the tangent line to the graph of your function at x=6. so, let's throw out the word "derivative" and substitute "slope of the tangent line to the graph at x=6."

Answer 29

ok nikin sorry for the confusion, yes the y value is -3/2

now to find the slope you also need to know how fast the graph changes at that point. Which can be done by using an x value close to the point (so off by 0.1 or 0.01 or 0.001 or whatever) and find the y value of that.

So let's try with 6.01

Answer 30

@mathmale 'derivative at point' (it's usually a way to prepare students for differentials)

Answer 31

So:

If we take the function f(x)=-9/x and substitute x=6 (as zimmah has already suggested), the function value, f(6)=-9/6 = -3/2. Again, zimmah is correct.

Answer 32

Okay so after we have that, what's next?

Answer 33

Now I'll ask zimmah to assume that x=6.1 and to evaluate the function at that value.

Why? because this will give you TWO points on the curve. We need only find the slope of the secant line connecting those two points to approximate the DERIVATIVE of the function at x=6.

Answer 34

Summary: if x=6, y=-3/2.
if x=6.1, y=?

Then we need to find the slope of the secant line connecting these two points. @nikin?

Answer 35

The slope [at a point] is defined by the change of y divided by the change of x at that point.

Therefore, like mathmale said we need to use two points. One of which is the point itself (so x=6) and another point close to it (we need a point close to it because we want to know the slope at that point).

So, to continue, we first need to know a second coordinate near {6,-3/2}
for example the coordinate {6.1,y}

Answer 36

Cool! Thank you. I've evaluated f(x)=-9/x at 6.1 and have obtained -1.475 as the value of y.

@nikin : Please, would you take these two points and find the slope of the line connecting them (the secant line)?

Answer 37

Once again: the two points are (6,-3/2) and (6.1,-1.475) (or (6.1,=90/61) )

Answer 38

@nikin: I'm assuming you're familiar with the formula for the slope of a straight line. Apply that formula here. it will give you the slope of the secant line connecting these two points, and, as such, will give you the approx. value of the "derivative" of f(x) at x=6.

Answer 39

.025/.1

Answer 40

that seems about right

Answer 41

Try multiplying both numerator and denominator by 1000, to simplify.

Answer 42

so it is 1/4

Answer 43

does that match any of the four possible answers?

Answer 44

Yes it does thank you so much!

Answer 45

Thank you very much for the medal, Nikin. Anything else left to discuss about this particular problem?

Answer 46

well the exact answer is 1/4 so i'd be amazed if it wouldn't

Answer 47

you're welcome

Answer 48

Once again, we're using a non-calculus approach to find the "derivative" of this function at x=6, even though technically we don't yet know how to differentiate or to find the derivative.

It's important to know that the slope of the secant line to the curve either is the "derivative" or is very close to it.

Answer 49

@zimmah: U heeft een geweldige partner in dit werk geweest. Dank u.