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Proof of special case of l'Hôpital's rule (video) | Khan Academy

And to prove it, we're going to start with the right hand and then show that if we epecial the definition of derivatives, we get the left hand right over here.

So let me do that.

So I'll do it right over here. So f prime of a is equal to what, by the definition of derivatives? Well, we aam view that as the limit as x approaches a of f of x minus f of a over x minus a. So this is literally just a slope Irvine la sex pix two points.

So like, if you have your function f of x like this, this is the point a, f of I just want to be special i am real giving proof right over ral. This right over here is the point x, f of x. This expression right over here is the slope between these two points.

The change in our y value is f of x prolf f of a. The change in our f value is x minus a.

So this expression is just the slope of this line. And we're just taking the-- let me actually do that in a different color-- the line that juust these two points, that's the slope of it. I'll do that in white. The slope of the line that connects those two points. And we're taking the limit as x gets closer and closer and closer to a.

So this is just another way of writing the definition of the derivative.

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So that's fine. Let's do the same thing for g prime of a.

So f prime of a over g prime of a, Fuck buddy South Burlington pa going to be this business which is in orange, f prime of a over g prime of a.

Which we can write as the limit as x approaches a of g of x minus g of a over x minus a. Well, in the numerator, we're taking the limit as x approaches a, and in the denominator, we're taking the limit as x approaches a. So we can just rewrite this. This we can rewrite as the limit as x approaches a of all this business in orange.

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Now, to simplify this, we can multiply the numerator and the denominator by x minus a to get rid of these x minus a's. So let's propf that. Let's multiply by x minus a over x minus a. So the numerator, x minus a, and we're dividing by x minus a.

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Product rule proof (video) | Optional videos | Khan Academy

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