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Find $ f'(a) $.

$ f(x) = x^{-2} $

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02:55

Daniel Jaimes

01:08

Carson Merrill

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 7

Derivatives and Rates of Change

Limits

Derivatives

Harvey Mudd College

University of Nottingham

Idaho State University

Boston College

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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00:40

Find $f^{\prime}(x)$.$…

01:10

Find $ f $.

$ f&qu…

00:52

Find $f^{\prime \prime}(x)…

02:48

Find $f^{\prime}(x)$ and $…

00:46

find $f^{\prime \prime}(2)…

02:49

Find $ f'(x) $ and $ …

00:36

So in this problem we are asked to find f prime at a when half of x is X to the -2. Well, by definition, if private A Is the limit as H approaches zero of F. Of a plus H minus F of a over H. All right. So that means that this is now the limit as h approaches zero F of a plus age and in this function F of X. So that is one over a plus H squared minus one over a squared all that is over eight. All right. So that means I have a limit as H goes to zero then of what? When I get a common denominator in the top. All right. And so I'll have a squared coming on top and then combine the fractions. That means I'll have a squared minus a squared plus two. A H plus H squared all over a squared times A plus H squared times H from the from the denominator previously. All right. So like this a squared minus at a square is but those two are gonna be gone, aren't they? So that means I have the limit as h goes to zero of minus to a H plus H squared over a squared times A plus h squared H. But I have an H in each term in the numerator so I can cancel one of those out. Can't I like that? So, that means I now have the limit As a joke goes to zero of minus to a plus H over a squared times A plus H squared. Okay, so taking now the limit as H goes to zero in the numerator, I'm just left with minus two way, aren't I? And the denominator. I have a squared times A squared, right? Because H goes to zero. So A plus H just goes to a square that let's just a square. So that means I'm left with -2 A over eight of the 4th. Well I can cancel 1, 1 of those AIDS out. and so that means I'm left with -2 over a cute is our derivative If prime it ain't.

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