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Hello.
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We're going to be evaluating the difference quotient for the given function.
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F of x equals x cubed as a quick reminder.
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The difference quotient helps us to determine the slope of any two points on our function x cubed.
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And as we move those points closer and closer together, we're getting closer to the exact value of the slope at one particular point on that function.
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And that comes in handy later on throughout your calculus journey.
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So for now, this is the process that we're going to follow.
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So the most confusing part of the difference quotient can be figuring out which values to substitute where into which function.
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So let me outline that first, um, here in our function f of x equals x cubed, we only have one x value.
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So it'll be pretty simple for us to substitute this a plus h right into our x value.
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So i'm gonna go ahead and do that below.
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Then i'm going to simplify before doing anything else with the difference quotient, just to make sure i have a simplified answer to move forward with.
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So if i use f of a plus, h equals a plus h cubed that again was just me plugging eight plus age into my x cubed.
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If i simplify that, then i can move forward with the rest of my difference question.
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So i will simplify that by first squaring a plus h so i'll have a squared plus h a plus a h plus h squared that could be further simplified to a squared plus two a h plus h square.
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Now i want to go ahead and multiply that by a plus h since i know originally i wanted to cube that function.
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So now we have a cute plus to a squared h plus a h squared plus a squared h plus to a age squared plus h cute.
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And remember what i'm left with right now is still on lee this first part of the difference quotient.
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So once we simplify this further that we can move forward with the rest of the difference quotient...