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title: Limit definition of the derivative

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Math 181 Miniproject 3: Texting Lesson.md
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Limit definition of the derivative
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<div><img class="left"/><div class="alert gray">

Hey, I missed the last class and was trying to catch up on notes. But I get so lost when I try to make sense of the limit definition of the derivative. Will you please help me?

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<div><div class="alert blue">

Of course, I can help you out! Do you know what the limit definition of the derivative is and is there a specific question I can help you with?

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<div><img class="left"/><div class="alert gray">

Yeah, the limit definition of the limit from what I understand is 
$f'(x)=lim_{h->0}(f(x+h)-f(x)))/(h)$.

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<div><img class="left"/><div class="alert gray">

So, for the function $2x^2$-5, how would I use the definition to find the derivative?

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<div><div class="alert blue">

Great question! To get the derivative the function has to be plugged in to the definition of the derivative. Like this:
$f'(x)=lim_{x->h}(2(x+h)^(2)

-2x^(2)-5)/(h)$

after distributing the terms and cancelling out terms we get:

f(x)=lim_{h->0}(4xh+2h^(2))/(h)

Then factor out the "h" from the numerator and cancel it out with the denominator.

$f'(x)=lim_{h->0} 4x+2h$

Finally when you apply the limit, you are left with the derivative. 
f'(x)= 4x

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<div><img class="left"/><div class="alert gray">

oh, that makes a lot more sense now. But why does this matter? Could you explain what the function and derivative would stand for in terms of the real world?  

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<div><div class="alert blue">

Oh yes! The derivative is the rate of change of the function.
So, for example let's say that the function $2x^2$-5 tells us the position of a moving car at a certain time. Can you guess what the derivative could tell us?


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<div><img class="left"/><div class="alert gray">

How fast the car is moving?

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<div><div class="alert blue">

Yes! Or the rate at which the car is moving from one position to another. The derivative we found, 4x, can tell us the rate in which the car is moving at a certain point in time. Not just in this case but for whichever situation the function stands for, the derivative will inform us the rate in which the function is changing. 

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