Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

While we have stated the chain rule, for the most part we examined the special case of the generalized power rule, as we have been dealing only with algebraic type functions. Later on in the text, we shall consider other kinds of functions which will utilize the chain rule. we anticipate other applications of the chain rule.Suppose $y=E(x)$ where $\frac{d}{d x}(E(x))=E(x),$ determine $\frac{d}{d x}\left(E\left(x^{2}\right)\right).$

$$2 x E\left(x^{2}\right)$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 6

The Chain Rule

Derivatives

Campbell University

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

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.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

02:31

While we have stated the c…

01:44

02:34

02:47

The generalized power rule…

00:24

02:21

Use a tree diagram to writ…

02:27

03:02

00:55

Let $w=f(x, y)$ be a funct…

01:55

Let $z=f(t) g(t) .$ Use th…

03:43

All right, so this is a problem or where were given the clue that the derivative of E to the E of X is equal to e of X. And if you're curious where this is, the chain bowl is. What we're seeing is that the derivative of this function is itself. And maybe I shouldn't underline everything but notice that X is a function, so it stays the same. Onda. Technically, what's happening is for this to be the changing role we're multiplying by the derivative of X, which is just times one. So as I moved to the second part of this where they're asking to find the derivative of e of X squared, basically what this boils down to is it's equal to itself. So I guess I should point out that my function is going to stay the same. But then what I have to do is multiplied by the derivative of the inside of the derivative of the inside is two X. Now, a lot of books would clean this up as two x b of X squared. This is your final answer. Now, if you're curious later on, you'll learn that the derivative off That's why I accidentally said it a couple times. E to the X is E to the X And then when you mix in the chain Bull, what we technically just found is the derivative of E to the X squared function. So it's two x e to the X squared, but I'm gonna way ahead of myself. This is the answer for today. I want to move on.

View More Answers From This Book

Find Another Textbook

04:06

Use the first derivative to determine where the given function is increasing…

03:44

Use the second derivative test to classify the critical points.$$f(x)=(x…

01:09

$$\text { Find } f^{\prime \prime}(x) \text { for. (a) } f(x)=\frac{\left(x^…

05:30

For the function $y=x /\left(x^{2}+1\right):$ (a) Find the derivative. (b) F…

07:08

In Exercises $29-38,$ for the function determined by the given equation (a) …

01:32

Sketch the graph of the function defined by the given equation.$$y=f(x)=…

02:46

$\$ 1250$ is deposited into an account for 10 years. Determine the accumulat…

02:40

A large cube of ice is melting uniformly at the rate of 6 cubic inches per s…

01:36

Determine where the function is concave upward and downward, and list all in…

02:09

Find the points on the circle $x^{2}+y^{2}=9$ (a) closest to and (b) farthes…