Find the slope, m, of the tangent line to the curve x²

Find the slope, m, of the tangent line to the curve x² - xy + y³ = 11 at the point (3, 2). a) m = -4/9 b) m = 5/4 c) m = -6/11 d) m = -16 e) m = -5/11 d/dx (x² - xy + y³) = d/dx (11) y = f(x) 2x - [1 · y + x · dy/dx] + 3y² · dy/dx = 0 2x - y - x dy/dx + 3y² dy/dx = 0 x = 3 y = 2 6 - 2 - 3 · dy/dx + 12 dy/dx = 0 9 dy/dx = -4 dy/dx = -4/9

Transcript

00:01 Okay, so you wanted me to help you explain this answer.

00:04 And i'm thinking that you're maybe not sure why we're taking the derivative in the first place.

00:09 So i want you to recall there's several equivalencies to what derivative means.

00:16 Or there's several things that taking the derivative gives us.

00:20 One of those things is rate of change.

00:23 If you ever see find the rate of change, that means take the derivative.

00:27 Another thing is acceleration.

00:28 If you ever see find the acceleration, that means take the derivative.

00:32 And finally, the last thing is find the slope of the tangent line at a specific point.

00:39 So if you see slope of tangent line, that means take derivative.

00:42 So as an example, over on the right, i tried to kind of loosely graph what this graph looks like.

00:50 You can plug it into an online graphing calculator if you would like.

00:54 Work.

00:54 And what we're doing is at point three comma two, which i tried to kind of graph with that blue dot there, we're looking at the tangent line.

01:03 So the tangent line is the line that basically touches our graph at that spot.

01:11 And so you can see that my line there is tilted down.

01:15 So it should end up having a negative slope, which we know a is the answer and it has a negative slip of 4 9ths, which roughly speaking in my hand drawing, that's what it is.

01:29 Okay, so now i'll go through the steps with you to hopefully get whatever your question was here.

01:36 So we're going to take the derivative of both sides, and i'm going to just use prime notation.

01:41 The previous person used d dx notation, same deal.

01:47 We're taking the derivative.

01:48 So, let's do each piece.

01:51 Derivative of x squared, well that's just the power rule, bring the 2 down and then take 1 away from the exponents, so it becomes 2x.

01:59 Now we're going to subtract the derivative of this next portion, but we have to remember that y is really a function of x.

02:10 So we really have two pieces here that have x in them.

02:14 So what do we do when we're taking the derivative of two things multiplied together? we have to use the product rule.

02:21 So we're going to use the product rule here.

02:24 So i'm going to put brackets for our answer, right, because product rule gives us two things added together.

02:30 So the product rule for this, we do the derivative of the first part, which is just x.

02:36 What's the derivative of x? well that's just 1 times the derivative of y.

02:42 Well, we don't know what the actual function of y is, you know, it's got a bunch of x's in it.

02:49 That's okay, we don't know what it is, so we can just notate it as y prime, the derivative, whatever it is, there it is.

02:56 Okay, so then for the second part of the product rule, sorry, no, we don't write it as y prime right there, right, product rule is take the derivative of the first part and then just write the second part...

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