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Find the instantaneous rate of change of $y$ with respect to $x$ at the given $x$ -value$$y=m x+b ; \quad x=a$$

$$m$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 7

Marginal Functions and Rates of Change

Derivatives

Missouri State University

Oregon State University

University of Nottingham

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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Find the instantaneous rat…

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Rate of Change Show that t…

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so to find the instantaneous rate of change for your basically in your function y equals MX plus B. You have you take your derivative of the function so D of life, which means your derivative of your whole function with respect to the variable X. So this is the variable you're using to calculate your derivations. So to do that, we look at each term so we'll start with MX, and I am. Here is a constant and X is the variable we're doing. So define the derivative of MX you take. Let's say it's MX 21 because it's the one that's just X. You multiply your constant by the X opponent and you subtract one from the exponents position. So the derivative of MX will be, um, this is not a divide. We'll just do that. And then you take your second very or your second term, and you calculate the derivative. So be here is a constant. And since there is no X in this term, the constant the derivative, a constant is just zero. So we get our derivative to be equal to M, and this is for any value. So here we have X equals a. But since there's no xperia ble in the derivative, the we know the rate of change is constant, which makes sense, because if you look at a linear function anywhere in the graph, you have a constant rate of change. This is the same as this, which is the same as that.

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