WEBVTT
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and this problem. We're finding the equation for the
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tangent line to a curve. Specifically, the function
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of that curve is log rhythmic, and we're also
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given a point at which we have to find the
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tangent line. So the function were given is why
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equals the natural log of X squared minus three x
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plus one. So the first thing that we need
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to do before we can find the tangent line is
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find the derivative. So we need Y prime.
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How are we going to do that? We have
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to use training role. So why prime would be
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equal to one over X squared minus three x plus
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one times two X minus three. Now, if
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you're confused about that, basically what you do with
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chain rule because we know the derivative of the natural
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log is one over X, so we would have
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one over X. But in this case, it's
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this entire input and then we'll multiply by the derivative
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of that input. And then we can simplify Why
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? Prime to be two X minus three over X
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squared minus three X plus one. So we have
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the derivatives, but we need the slope of the
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tangent mind at the 0.30 so we can find why
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, prime of three, why crime of three would
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be equal to sticks minus 3/9 minus nine plus one
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so it would simplify to three. And now we're
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very close to finding the equation for a tangent line
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. We can use this point slope form why,
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minus why not equals m times X minus x dot
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So we can just plug in the point that were
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given why my zero equals three times X minus three
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. So you can essentially write the tangent line in
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this form or you can simplify it in point in
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slope form and we would have Why equals to reacts
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minus nine. So either way, they both mean
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the same thing. These air both the equation for
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the tangent line to our curve at that point.
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So I hope that this problem helped you understand a
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little bit more about how we confined the equation for
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a tangent line, using our knowledge of differentiation of
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logarithmic functions and then plugging in the point in our
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, um, coordinate plain to find the overall tangent
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line