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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 $y=L(x)$ where $\frac{d}{d x}(L(x))=\frac{1}{x},$ determine $\frac{d}{d x}\left(L\left(x^{2}+1\right)\right).$

$$\frac{2 x}{x^{2}+1}$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 6

The Chain Rule

Derivatives

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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.

01:49

While we have stated the c…

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02:34

02:47

The generalized power rule…

00:24

00:33

Use Version 1 of the Chain…

03:14

Using the Chain Rule, show…

02:27

Use a tree diagram to writ…

03:02

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00:55

Let $w=f(x, y)$ be a funct…

Chain Rule for powers Use …

All right. So the way I understand this problem is that we're starting with the derivative of some function l affects is equal to one over X eso when we get to the directions where they're asking you to identify the derivative of so d d x of el of X squared plus one. Uh, the way I would think think of this problem is that Well, if I go back to this problem, let me let me explain. The chain role for a second is X is representing a function. So the function stays in the denominator. And then you take the derivative of X. Well, the derivative of X would be times one on that's where the chain was coming in. So as I'm looking at the answer for this one, I'm thinking Okay, well, the function is going to stay the same, so it's gonna be one over X squared plus one. But then you have to multiply by the derivative of the inside. Hence the changeable, which would be times two x now, as faras multiplying fractions, this two X is just going to go into the numerator. So we're looking at two x times one is two x over X squared, plus one. If anybody is curious later on in the in the text, what you'll figure out is this is the natural log function. So maybe it's curious. The derivative of natural log of X is equal to one over X s. Oh, this. This is a precursor for the natural log function. But anyway, here's your answer.

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