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If $y=\left(3 x^{2}+9\right)^{3 / 2},$ find the rate of change of $y$ with respect to (a) $x$(b) $\frac{3 x}{x^{2}-1}$

(a) $3 x^{2}$(b) $x / 4$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 7

Marginal Functions and Rates of Change

Derivatives

Missouri State University

Oregon State University

Harvey Mudd College

Idaho State University

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.

01:47

. Find the rate of change…

01:12

Find the rate of change of…

So in this question we have to find the rate of change of Y. With respect to act so we can ride this bike you into So you equal U. Equals three X. To the power one by two. And we will allen two X. So divide by the X equals you into divide the X plus we into the U. S. D. X. So three X to the power one by two into differentiation of Ln two X plus V. Ln two X into differentiation of three X. To the power one by two. Equal equal okay three X three. Power 1 x two into 1 x two. X. into two plus Ellen two X two in two. Three by two X. To the power minus one by two. So equal equal three extra. The power one by two minus one less. Ln two X into three by two X. To the power minus one by two. So take three X to the power minus one by two. Common one plus one by two. Ellen two X. So Dubai bye. The x equals three by route one plus Yeah, one by two. Ellen two

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