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Use the alternate form of the derivative given in Exercise $37,$ to compute $f^{\prime}(x)$ for the function defined in: (a) Exercise 21; (b) Exercise 22; (c) Exercise 23; (d) Exercise 24.

(a) 0(b) $m$(c) $-3 / x^{2}$(d) $2 / x^{2}$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 1

Slope of a Curve

Derivatives

Harvey Mudd College

Baylor 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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02:31

we use this definition to compute the derivative of the following functions 1st. FX. Eco's 53. So the relative of this function is just limits. X two goes to eggs. Yeah, Functioning value is 53 miners. 50's great. And hence the limited of 80 That is the derivative of a constant function zero. Exercise 22. It just substitute fx with the expression of the function Which is MX two Plus B miners and X. Class B. Over the next two minor setbacks which he calls M. X. two miners and the eggs Over X. two minor setbacks. Sorry, this should have a limit and which is just name is X two goes to acts am time six to minor sacks over acts to minors access which equals air And for Exercise 23. Yeah. Prime eggs echoes Limit X two goes to eggs. The subsidy to the effects with the expression that is three over X. to minors spray over X over X two minus X. We want to eliminate the denominator and hence we just multiply that Numerator by X. two time next which becomes three times Eriks miners three times x two over at times next to The over the next two times x. Uh Sorry you should be a limit here Which equals name it. X two goes to X. My nursery times X two minus X times it's times yaks too. And that's true council. And this becomes limited Next to those two eggs. My nursery X times X two. And we just substitute the X two is air which cares Miners three over x squared For for Exercise 24. The process is similar. We substitute FX visits expression which he calls miners too. Over access to Miners, Miners two Over X. And the denominator is X. two minor setbacks. Rich Nichols two over X -2 racks too. Wow. There should be a limit here. And this equals limit X two point goes to acts and they multiply the numerator by ads times X two. This becomes too next to minors to eric's thanks. Times two he called, he calls limit X two goes to Arabs and this to cancel. So there's, he calls limits access to goes to act to over Eriks times next to. So here we just substitute X two ways X and this is two over X squared.

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