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• So I've been requested to do the proof of the derivative of

• the square root of x, so I thought I would do a quick

• video on the proof of the derivative of the

• square root of x.

• So we know from the definition of a derivative that the

• derivative of the function square root of x, that is equal

• to-- let me switch colors, just for a variety-- that's equal to

• the limit as delta x approaches 0.

• And you know, some people say h approaches 0,

• or d approaches 0.

• I just use delta x.

• So the change in x over 0.

• And then we say f of x plus delta x, so in this

• case this is f of x.

• So it's the square root of x plus delta x minus f of x,

• which in this case it's square root of x.

• All of that over the change in x, over delta x.

• Right now when I look at that, there's not much simplification

• I can do to make this come out with something meaningful.

• I'm going to multiply the numerator and the denominator

• by the conjugate of the numerator is what

• I mean by that.

• Let me rewrite it.

• Limit is delta x approaching 0-- I'm just rewriting

• what I have here.

• So I said the square root of x plus delta x minus

• square root of x.

• All of that over delta x.

• And I'm going to multiply that-- after switching colors--

• times square root of x plus delta x plus the square root of

• x, over the square root of x plus delta x plus the

• square root of x.

• This is just 1, so I could of course multiply that times-- if

• we assume that x and delta x aren't both 0, this is a

• defined number and this will be 1.

• And we can do that.

• This is 1/1, we're just multiplying it times this

• equation, and we get limit as delta x approaches 0.

• This is a minus b times a plus b.

• Let me do little aside here.

• Let me say a plus b times a minus b is equal to a

• squared minus b squared.

• So this is a plus b times a minus b.

• So it's going to be equal to a squared.

• So what's this quantity squared or this quantity squared,

• either one, these are my a's.

• Well it's just going to be x plus delta x.

• So we get x plus delta x.

• And then what's b squared?

• So minus square root of x is b in this analogy.

• So square root of x squared is just x.

• And all of that over delta x times square root of x

• plus delta x plus the square root of x.

• Let's see what simplification we can do.

• Well we have an x and then a minus x, so those

• cancel out. x minus x.

• And then we're left in the numerator and the denominator,

• all we have is a delta x here and a delta x here, so let's

• divide the numerator and the denominator by delta x.

• So this goes to 1, this goes to 1.

• And so this equals the limit-- I'll write smaller, because I'm

• running out of space-- limit as delta x approaches 0 of 1 over.

• And of course we can only do this assuming that delta--

• well, we're dividing by delta x to begin with, so we know

• it's not 0, it's just approaching zero.

• So we get square root of x plus delta x plus

• the square root of x.

• And now we can just directly take the limit

• as it approaches 0.

• We can just set delta x as equal to 0.

• That's what it's approaching.

• So then that equals one over the square root of x.

• Right, delta x is 0, so we can ignore that.

• We could take the limit all the way to 0.

• And then this is of course just a square root of x here plus

• the square root of x, and that equals 1 over

• 2 square root of x.

• And that equals 1/2x to the negative 1/2.

• So we just proved that x to the 1/2 power, the derivative of it

• is 1/2x to the negative 1/2, and so it is consistent with

• the general property that the derivative of-- oh I don't

• know-- the derivative of x to the n is equal to nx to the n

• minus 1, even in this case where the n was 1/2.

• Well hopefully that's satisfying.

• I didn't prove it for all fractions but this is a start.

• This is a common one you see, square root of x, and

• it's hopefully not too complicated for proof.

• I will see you in future videos.

So I've been requested to do the proof of the derivative of

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# Proof: d/dx(sqrt(x)) | Taking derivatives | Differential Calculus | Khan Academy

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yukang920108 posted on 2022/07/12
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