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Find the instantaneous rate of change of $y$ with respect to $x$ at the given $x$ -value$$y=3 x+2 ; \quad x=5$$

$$3$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 7

Marginal Functions and Rates of Change

Derivatives

Missouri State University

Campbell University

Oregon State University

University of Michigan - Ann Arbor

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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01:49

Estimate the instantaneous…

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01:40

So to find the instantaneous rate of change, you take the derivative of your function, and then you plug in your X value to solve. So to find a derivative of a linear function, it's fairly simple. You have to find the derivative of each term with, or so the derivative, the whole function with respect to X. That's what this term means. So to find the derivative, a linear function, you just go to each term. So the derivative of three X you take your how many ex very or the power of your ex, which is three X, the one and you subtract one from the Explain it and you'll multiply 12 the constant in front of it. So one minus one is zero and X to the zero is just one. And so your derivative for this term is three, and then two is a constant, and it doesn't have any X terms attached to it, so that the derivative of that term is just zero. So here are derivative is three, which means that no matter what the X variable here is, it could be any valuable and that variable. And in this case, it's five, your rate of change will be three, Which makes sense because if you look at a linear slope you have or a linear function, your rate of change is constant, no matter where you are in the ground.

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