Question

for f(x) = 1/x, find f '(2) by using limits. can anyone help?

Answer 1

Do you know the definition of a derivative?

Answer 2

yes

Answer 3

So when you use it, what do you get? Where are you stuck?

Answer 4

im not sure how to even begin to start this problem. im not sure where the use limits to solve comes in

Answer 5

Start with the definition of a derivative. Just start using it as best as you can, don't worry about messing up, just try it out and see how far you get with it. Remember, definition of a derivative looks like this: \[f'(x)=\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\] and your f(x)=1/x

After you find the derivative, then you can plug in 2 to find f'(2)

Answer 6

so your saying find derivative of 1/x

Answer 7

Exactly.

Answer 8

if so the derivative is -1/x^2

Answer 9

Sure, but did you take the derivative using limits?

Answer 10

no i just took the derivative of 1/x. not sure how to use the formula that you gave.

Answer 11

What are you unsure of? I can't help you unless you try it out and show me where you fail. Otherwise I will just keep telling you to try it out and you'll get nowhere.

Answer 12

on the formula you gave what is h what is x

Answer 13

h is the value that we're taking the limit of as it approaches 0. Remember the slope formula from algebra? \[m=\frac{y_2-y_1}{x_2-x_1}\] The derivative is exactly the same thing as this, only we're adding a little extra to it. What if point 2 and point 1 were right next to each other? That's what calculus is about since at any point on a a curve that's not a line we have a different slope anywhere.

h is just the distance between points like this: So let it sorta sink in, see how we picked h to be an arbitrary distance between the point x and (x+h)? Then we evaluated our function f(x) at x and x+h to get our "y" value f(x) and f(x+h), so those are really our y1 and y2 values.

The slope between those two points is not very good though, it's kind of the slope, but not really. But as you make h get closer and closer to zero you start to have just the slope at the point x. That's how you get a derivative basically.

So in this case, f(x)=1/x and f(x+h) = 1/(x+h) right?