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Find all points at which the derivative of $y=\left(x^{2}-3 x-4\right)^{2}$ is equal to zero. [Hint: To find the derivative, remember that $\left.A^{2}=A \cdot A .\right]$

$(-1,0),(3 / 2,625 / 16),(4,0)$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 5

Derivative Rules 2

Derivatives

Campbell University

Oregon State 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.

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.

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All right. So this problem is getting you to practice the derivative and setting it equal to zero minus three X minus four squared s. I'll just call this why, But what they wanted you to do is recognize that this is equal to multiplied by itself. So then what you can do is the product rule where the derivative de y DX is equal. To take the drift of of the left side, which is two X minus three. Leave the right side alone squared minutes three X minus four and then plus the well, I'll take the door of the right side. So the two x minus three and leave the left side of them Uh, X squared minus three explosive minus four. But as you notice in that problem, um, they're exactly the same. This piece and this piece are exactly the same. So what that means is that there's too of these pieces to x minus three and this X squared minus three X minus four. Eso what you could do since the direction say, find where the drift of equal zero is. You can quickly use the zero product property and say, Oh, well, if X equals three halves. Um, you know, then two times three has to be three ministries. Zero. I don't care what anything else is that's going to give me a zero or this X squared minus three x minus four could equal zero. I don't need that parentheses there. Sorry about that. But anyway, as you factor, that would be negative for plus one. So X minus forward X plus one eyes equal to zero. So that tells me by the zero product property that X could be four or X could be a negative one. Now the directions do also ask for the ordered Paris says at what points? What's nice about these two points is thes air. Also zeros in the original problem. Um, so what that means is that when excess for the Y value of zero and X is negative one the y value of zero. That's not the case I put in boxer on these two answers. For this one, you need to plug in three halves into the original problem, and you get 625 16th. I used a calculator, uh, 34 squared minus three times three has I should have said three have squared minus three times three has minus four, which should be 25 force because then you square it and you'll get 625 16th eso the box dancers down here what we're looking for because they're ordered pairs 123 points.

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