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Suppose $ y = f(x) $ is a curve that always lies above the $ x $-axis and never has a horizontal tangent, where $ f $ is differentiable everywhere. For what value of $ y $ is the rate of change of $ y^5 $ with respect to $ x $ eighty times the rate of change of $ y $ with respect to $ x? $

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$y=2$

00:45

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 4

The Chain Rule

Derivatives

Differentiation

Campbell University

Baylor University

Idaho State University

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.

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in this problem, we have a description of some function y equals f of X, and then we have some other function which I'm going to call G and it is white of the fifth Power. So that would be f of X to the fifth power. And then we have a description of the relationship between the derivative of G and the derivative of why So we want to find the point where the derivative of G is 80 times the derivative of why so we could write it like this. The derivative of G would be g prime of X and we want that to be 80 times the derivative of why, which would be f prime of X. So let's go ahead and find the derivative of G. So notice that G of X is a composite function. So we use the chain rule to get its derivative would bring down the five and then we raised F of X to the fourth. And then we multiply by the derivative of the inside, which is F prime of X. So let's substitute that for G Prime of X. Now notice we have a factor of F prime of X on both sides so we can cancel that. So now we're left with five times f of X to the fourth equals 80. Let's divide both sides by five so f of X to the fourth equal 16. Let's take the fourth root of both sides and we know that we're looking for a positive because we're told that f of X lies above the X axis so we don't want a negative value just positive. So f of X equals two. So what's the value of why we just found it? F of X means why? So why equals two at this point where the derivative is 80 times the original derivative?

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