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(a) If $ f(x) = (x^3 - x)e^x, $ find $ f' (x). $(b) Check to see that your answer to part (a) is reasonable by comparing the graphs of $ f $ and $ f'. $

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(a) $e^{x}\left(x^{3}+3 x^{2}-x-1\right)$(b) See explanation for result

00:31

Frank Lin

02:19

Clarissa Noh

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 2

The Product and Quotient Rules

Derivatives

Differentiation

Missouri State University

Campbell University

Idaho State University

Boston College

Lectures

04:40

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.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

02:39

$\begin{array}{l}{\text { …

01:31

(a) If $f(x)=\left(x^{3}-x…

06:50

(a) If $f(x)=x+1 / x,$ fin…

02:15

02:21

(a) If $ f(x) = (x^2 - 1) …

05:19

(a) If $ f(x) = x + 1/x $,…

02:38

01:07

(a) If $f(x)=x^{4}+2 x,$ f…

07:55

(a) If f(x) = x2 ? 1/x…

All right, so our first part of our problem part a is to find the derivative of f of x. So, let's write down f of x f of x is defined as x, cube minus x in parenthesis times e to the notice we have a product here. So i'm going to call this function: u and this function v and what we're going to do is apply the product rule and the product rule basically says. If i want to take the derivative with respect to x of function- u times v, then that equals derivative of? U times v plus derivative of v times? U, so let's go ahead and apply that okay, so f, prime of x, then will be? U prime! So that's a by power rule will get 3 x squared minus 1 times e to the x, because that is v. They we're gonna, multiply or add and then use the prime. The prime is derivative e to the essex is itself, that's super handy and then u is just x, cubed minus x. So that is our derivative. We can clean it up a little bit by going ahead and factoring out to the x because notice both terms have e to the x and then, let's see what's left over, we have an x cubed and let's go ahead and put and write or a typical Descending power order, so then we have plus 3 x squared. We have a minus x from there and a minus 1 from the left. Okay. So that is our solution of part. A we found our derivative, then part b. We are asked to graph and see if our solution makes sense. Let'S take a look o, so basically we have went ahead and graph. These we just need to identify which ones which okay, the 1 that goes through 00, would be f of x. So the blue 1 is f of x and the red 1 is a prime of x and i'm going to make this look a little bit better just so we can tell which 1, which okay, so this is f of x. Okay, let's just check a few points to see if it makes sense. Okay, so notice that f of x has a minimum right here, and that means the f prime of x should have a critical point and in fact it should go from a negative to a positive and it does so that matches. Let'S look at f of x. F of x has a max about there, which means f, prime of x, should have a critical point which it does right there and it should go from positive to negative for the map, so everything's, looking really good. Let'S see, there's 1 more minimum about here. So remember this was a mine and this was a max for the f of x. Then we have a men again and notice. A prime of x has a critical point and it is going from negative to positive. So looking really good that are derivative worked and it matched our graph. So excellent have a wonderful day and see your next t.

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