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Determine the derivative at the given point on the curve using equation (2).$f(x)=3-2 x-x^{2}$ at the point (-1,4).

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Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 1

Slope of a Curve

Derivatives

Baylor University

University of Michigan - Ann Arbor

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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Okay. The question is, what is the derivative of dysfunction? Sorry. The slope of that function at the point minus one. So the function is three minus two x mhm minus X square. And the point is, one is one and four. So if primer bags is defined, that f of X plus age So this is three minus two times X plus age minus X plus age square. Okay, this is if off x plus age minus half of eggs. So you put the whole thing in the bracket here. So three minus two eggs. Why this x square And the whole thing they wanted by age in each goes to zero. If you expand this, so you're going to cancel it straight. You have three years minus three year. You're going to get there minus. Hey, you're gonna have a minus two X here, right, minus two times X behavior plus two X because we have a minus. Another minus is to cancel each other. And this excess square here, plus excess square cancels the minus Excuse Quit from there their term. So you're going tohave minus two age, right, and then minus two x h and minus h A square. Okay. And the whole thing divided by age when the h goes to zero. So this is going to be minus two minus two x on B minus age in the age was 20 so the age drops out. Okay, so this was a friend of eggs. So if we know X becomes minus two minus two x. Okay. And at point minus one. So a brain one minus one becomes minus two plus two. She's the zero so derivative of this function. Ah, point minus one is equal to zero.

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