Two+Definitions+of+derivatives


 * "LETS DO THIS SHIZ" -** Archimedes



=Dos Hermanos, Un Camino=

Now there are two ways of defining derivatives. Both ways are just as BAMF as these gentlemen right here.


 * f'(c)**, as you've already learned, is the slope of the curve at a certain point, and you might be thinking you're getting to know this baby pretty well.


 * f'(x)** seems to be pretty popular around these parts too, looking all fancy and plotted and cool.

But hold on there partner. You seem a little confused. "I'm still a little shaky on how they fit together."

Have no fear. Mama RoRo's right over here.

Are you ready to become awesome?



Thought so. Here we go.

=It's Clobberin' Time=  THE DEFINITION OF THE DERIVATIVE AT A POINT (assuming the function is differentiable & limit exists) First off, we can estimate a derivative of a point on a function by finding a change in f(x) and dividing it by a corresponding change in x. This is called finding a difference quotient. As the change in x and the change in f(x) decrease, the difference quotient becomes more and more accurate at estimating the derivative at that instant since the two points being compared have a smaller difference, i.e, they are closer to each other on a graph. Evaluating a derivative at x=c is done by finding the instantaneous rate of change of f(x) with respect to x at x=c, shown below with limit notation that implies that the change in f(x) and change in x approach zero: [|Also note, finding the derivative at the point x+c is the same as finding the slope of the tangent line to the graph at that point]. The fact that the line is tangent to the graph is a graphical interpretation of the meaning of derivative. You learned about this property in the section "Graphical Representation".

__Example #1__  THE SECOND DEFINITION OF THE DERIVATIVE AT A POINT (assuming the function is differentiable & limit exists)  
 * Δ//x// || = || //h// ||
 * Δ//y// || = || //f// (//x// + //h//) − //f// (//x//) ||
 * Δ//y// || = || //f// (//x// + //h//) − //f// (//x//) ||

You may be wondering what all of that stuff is and what it means. :o :$ :/

Earlier, we explained that the graphical equivalent of the derivative at a point was the slope of the line //tangent// to that point. As seen below at point x=P.  Let's now assume we have two different points on a graph, points P and Q with which we must evaluate the derivative at P. To evaluate the derivative at point P, the change in the x and y axes between point P and point Q must lessen. We are bringing Q closer and closer to P and finding the slope between the two points because eventually, the slope between the points will become the slope of the point P. Thus, the secant line becomes the tangent line we found before (but by another method now). You can see this in the 2nd figure below.

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Now lets tie this all together.

By expressing the derivative at point P as it relates to a change in x, delta x, we can better express and generalize the derivative at the point P. Mathematicians like this definition more, and there's a pretty good reason why. With this second derivative definition, we can now derive power functions!

Example #2 Given, find f'(x).

If, then. Restriction: The exponent n is a constant.
 * __Property: Derivative of the Power Function__**

On a side note: __Using TI-83 to Calculate Derivatives__ To find the derivative function: - enter original equation in Y1 - press Math+8+Vars+->+1+1+,+x+,+x+)+graph To find the derivative at a point: - after following above steps - press 2nd+trace+1+(your c value)

What can we learn from two definitions of derivatives? Can we evaluate the exact derivative at a point? Can we use limits to evaluate the derivative? Will this be useful to us later? :l **If** you said //yes// to all these questions**,**

=.=

Moving right along, lets see some derivative properties!


 * __Properties: Differentiation__**

Derivative of a Sum of Two Functions Derivative of a Constant Times a Function (k is a constant) Derivative of a Constant (C is a constant)

If y=f(x), instead of writing f'(x) you can write any of the following: y', read "y prime" (a short form of f'(x)) , read "dy,dx" (a single symbol, not a fraction) , read "d, dx of y" (an operation done on y)
 * __Terminology: dy/dx and y'__**

// If you're sick of math and want to take a break, go ahead and click the link below. Turn up your speakers and prepare to follow your dreams.

// =Your reward for being so good about all of this:=



media type="custom" key="5830923" Sources: http://www.math.hmc.edu/calculus/tutorials/limit_definition/ http://www.themathpage.com/acalc/derivative.htm#definition http://www.tutorvista.com/ks/definition-of-derivative-at-a-point http://mathworld.wolfram.com/Derivative.html [|__http://www.teacherschoice.com.au/Maths_Library/Calculus/plot_derivative_graphs.htm__] [|__http://webpages.math.luc.edu/~ajs/courses/131fall2004/solutions/test2.html__] [|FOERSTER, PAUL. //CALCULUS CONCEPTS AND APPLICATIONS//. 2004-05-31, 2004. Print.]



Topic: Two Definitions of Derivatives Group Members: Nico and Veronica

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Love the creativity and use of media. Engaging wiki page. Great explanations and examples that are supported by images. Good idea including calculator instructions as well. Great Wiki Page!