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Derive equation (4).

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 5

Applications II - Business and Economic Optimization Problems

Derivatives

Baylor University

University of Michigan - Ann Arbor

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.

01:46

Derive formula ( 3 ).

02:17

Derive formula ( 8 ).

So, to derive formula Three, we're going to be using some and difference identities from earlier in this chapter in order to turn the right hand side into the left hand side. So this right here is the right hand side, and so we're going to apply some some difference identities to it. So sign of Alfa plus beta, we know from our formula is sign Alfa Co sign beta less co sign Alfa sign beta and then sign of Alfa minus beta equals sign Alfa Co sign Beta minus co sign Alfa Sign Beta. So what we noticed here is that we have two sets of life terms. So we have sign Alfa Coast Beta twice, and we also have Who's Alfa signed? Big device. So I mean this all together we get 1/2. We have two copies of science Alfa Co sign beta and coastline Elfin sign. Beta actually gets canceled out because we have a plus and a minus. So that term goes away. And so now we cancel out the 1/2 and the to to get sign Alfa Co sign Beta, and this is the left hand side of the equation that we want it. So this is our stopping point. We have now proven this formula, so we are done

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