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Find the instantaneous rate of change of $y$ with respect to $x$ at the given $x$ -value$$y=\left(x^{2}-1\right)^{2}\left(x^{2}-3 x+1\right)^{3} \quad x=0$$

$$-9$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 7

Marginal Functions and Rates of Change

Derivatives

Campbell University

Baylor 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.

01:46

Find the instantaneous rat…

01:34

02:11

02:05

Yeah. So to find the instantaneous rate of change, we take the derivative of function y with respect to the variable X. So that's what this term means. The derivative of y with respect to X. And we do this by calculating the Drew two for individual terms. So we have three x squared. And so the derivative three x squared. You multiply the exponents to the number in front of the X, and then you subtract one from the exponent position. So the derivative three x squared is six X. Next we go to negative seven x. And so here we there's one in the exponents because X to the one X. And so you multiply that here and you subtract one. So that would give us negative seven and one minus. One is zero, and next to the zero is one, and next we have 22 is a constant, which means it doesn't have a variable in the same term as it in a conservative of a constant is zero. So, technically it's plus zero. So are derivative function. Here is six x minus seven. So to find the instantaneous rate of change at point X equals two, you plug in x two this derivative function. So we'll do that down here. So D Y d e x of two will be six times two minus seven, which is equal to five. Yeah, okay.

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