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Use the Chain Rule and the Product Rule to give an alternative proof of the Quotient Rule.[ $ Hint: $ Write $ f(x)/ g(x) = f(x)[g(x)]^{-1}.] $

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00:58

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 4

The Chain Rule

Derivatives

Differentiation

Missouri State University

Harvey Mudd College

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.

03:10

Use the Chain Rule and the…

01:11

Give a second proof of the…

02:39

Use the quotient rule to s…

01:38

Use the fact that $f(x)=u(…

02:15

All right, So we're going to find an alternative of the quotient rule our way to prove the quotient rule by taking the derivative of a product and using the chain rule. And so what we're aiming for is the derivative of a quotient. And so what we're going to do is take the derivative of this product instead. So according to the product rule, we have the first f of x times, the derivative of the second. Now, when we go to take the derivative of the second, we have to use the chain rule. So we're going to bring down the negative one, and then we're going to raise G of X to the negative second power, and then we're going to multiply by the derivative of G of X, which is G prime of X. Okay, so so far, we have the first times the derivative of the second. Now we're going to have plus the second, which was G of X, the negative one times the derivative of the first. And that would be the derivative of F. Okay, now our goal is just to manipulate this and change how it looks until it looks like the quotient rule. So let's notice that we have some negative exponents so we can turn some things into fractions. So the whole first term here is going to be the opposite of F of X times G prime of X over G of X squared. And that's looking pretty good because that looks a lot like our quotient rule, at least part of it. And then this whole second term here because we have a negative exponents, we can write that as f of X over g of X. Now, we have different denominators in these fractions. And if we want a common denominator, we should multiply the second fraction by G of X over G of X. Okay, that's going to give us our denominator of g of x quantity squared, which is what we want for the quotient rule and for the numerator. What I'm going to do, IHS Well, I'm gonna go back and write my prime sign on that F that f prime of X there it lost its prime Simon's prime sign in translation. Okay, now for the numerator. So I'm just going to write this part first, so that would be G of X Times f prime of X and then I'm going to write this part second. So I have minus f of x times g prime of X and this is the quotient rule.

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