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Problem

Find $ f'(a) $. $ f(t) = \dfrac{2t + 1}{t + 3}…

01:08

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Problem 32 Medium Difficulty

Find $ f'(a) $.

$ f(t) = 2t^3 + t $


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03:08

Daniel Jaimes

Related Courses

Calculus 1 / AB

Calculus: Early Transcendentals

Chapter 2

Limits and Derivatives

Section 7

Derivatives and Rates of Change

Related Topics

Limits

Derivatives

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Top Calculus 1 / AB Educators
Grace He
Catherine Ross

Missouri State University

Heather Zimmers

Oregon State University

Samuel Hannah

University of Nottingham

Calculus 1 / AB Courses

Lectures

Video Thumbnail

04:40

Limits - Intro

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.

Video Thumbnail

04:40

Derivatives - Intro

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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Watch More Solved Questions in Chapter 2

Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
Problem 29
Problem 30
Problem 31
Problem 32
Problem 33
Problem 34
Problem 35
Problem 36
Problem 37
Problem 38
Problem 39
Problem 40
Problem 41
Problem 42
Problem 43
Problem 44
Problem 45
Problem 46
Problem 47
Problem 48
Problem 49
Problem 50
Problem 51
Problem 52
Problem 53
Problem 54
Problem 55
Problem 56
Problem 57
Problem 58
Problem 59
Problem 60
Problem 61

Video Transcript

So in this problem we are given the function F. Of X equals two x cubed plus X. We were asked to determine if prime at a. Okay well by definition F. Prime at A. Is the limit As a church goes to zero of F. Of a plus H minus F. Of a. All over H. Okay so first of all we got to figure out half of a plus H. Well that's two times A. Plus H cubed plus A plus H. So that's two times E cubed was three A squared H. Plus three A. H squared plus one O. C. M. P. H. Sorry And that's plus A. Plus H. And so this becomes, well I have to a cute and I have three A squared H. And I have three A. That's not three. Sorry that's what babies and this one This will be six and that would be six. And then plus and a. Okay this is H cubed over here was and each plus an h cubed wouldn't it? Okay and let me just make this a little clear. Sorry about that. Just make this clear. That's too H cubed plus a. We'll see each. Okay F it a is simply to a cubed was hey isn't it? Okay so that means that F. At A plus h minus F. A. R. Numerator up here. Right our numerator up here. Well let's see The two A Cubes are going to cancel out. I'm going to have six A squared H. Plus six A. H plus two H cubed. Hey minus A. That will be gone. So I'll be plus H. Left on the end and when it, okay so that means now that I'm left with F. Prime at A. Is the limit as H. goes to zero Of six A. Squared H. Plus six h. plus two H. Cubed plus H. All over H. And now what do I see that age cancels? And that one cancels one of those cancels and that cancels doesn't it? Okay so yes I'm now left with this is the limit His age goes to zero of six A squared plus six A. Plus two eight squared. Well not mess up up here something didn't work out right I left two squared off of that turn right there didn't I? So that one squared. So this one was squared and so it should have been and H. Left there. Let's fix that up. This is the age squared cancel one of those out. Okay So this is six. Let's fix this up. Six A. Aah plus two H. Squared. Um So when I canceled this All right here that would have left or one. Okay so this is plus one down here. All right so now H. Goes to zero. So a return has an ancient it goes to zero. This one goes to zero and that one goes to zero So much with six A. Squared plus one is F. Prime today

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Related Topics

Limits

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Top Calculus 1 / AB Educators
Grace He

Numerade Educator

Catherine Ross

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Heather Zimmers

Oregon State University

Samuel Hannah

University of Nottingham

Calculus 1 / AB Courses

Lectures

Video Thumbnail

04:40

Limits - Intro

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.

Video Thumbnail

04:40

Derivatives - Intro

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.

Join Course
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