WEBVTT
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we're going to find the equation of the tangent line
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to the curve y equals square root one plus X
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cubed at the 10.23 The slope of the tangent line
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will be the derivative at that point. So let's
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start by finding the derivative. And I'm going to
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rewrite why, as one plus X cubed to the
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1/2 power. And so then I can use the
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chain rule, starting with the derivative of the 1/2
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power function. Bring down the 1/2 and raise one
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plus X cubed to the negative 1/2 power. And
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then we multiply that by the derivative of the inside
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, the derivative of one plus x cubed is three
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x squared. Now we're evaluating this derivative at X
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equals two. We could simplify first, or we
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could substitute the two in first and simplify after.
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So that's what I'm going to do. So 1/2
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times one plus two cubed to the negative 1/2 times
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three times two squared. Okay, so this gives
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us 1/2 times nine to the negative 1/2 times 12
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and nine to the negative. 1/2 is 1/3 so
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we have 1/2 times 1/3 times 12 So that is
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just too. So are slope is, too.
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Now we can use point slope form. Why minus
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y one equals m times X minus X one.
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To find the equation of the line where the point
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x one y one is 23 So we have why
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minus three equals air slope to times the quantity X
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minus two and we can simplify that and we'll have
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the equation of the line. So let's distribute the
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two. So why minus three equals two X minus
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four and then at three to both sides and we
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have y equals two x minus one.