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So we've got the equation x^2 + y^2 = 1,

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I guess we could call it a relationship.

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And if we were to graph all the points x and y that satisfy this relationship

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we get a unit circle like this.

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And what I'm curious about in this video is

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how we can figure out the slope of the tangent line

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at any point of this unit circle.

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And what immediately might be jumping out in your brain

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is well, a circle defined this way, this isn't a function

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it's not y explicitly defined as a function of x

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for any x value you actually have 2 possible y's

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that satisfy this relationship right over here.

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So you might be tempted to maybe split this up

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into two separate functions of x.

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You could say y is equal to the positive square root of 1-x^2

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and you could say y is equal to the negative square root of 1-x^2

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take the derivatives of each of these separately

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and you would be able to find the derivative

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for any x, or the derivative of the slope

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of the tangent line at any point.

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But what I want to do in this video,

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is literally leverage the chain rule

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to take the derivative implicitly

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so that I don't have to explicitly define y as a function of x either way.

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And the way we do that is literally just apply

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the derivative operator to both sides of this equation

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and then apply what we know about the chain rule.

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Because we're not explicitly defining y as a function of x

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and explicitly getting y = f'(x)

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they call this,

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which is really just an application of the chain rule,

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we call it implicit differentiation - Implicit Differentiation.

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And what I want you to keep in the back of your mind

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the entire time, is it is just an application of the chain rule.