1 00:00:00,012 --> 00:00:09,144 , Sometimes you don't have a function, you have a relation between two variables. A 2 00:00:09,287 --> 00:00:17,120 classic example is x squared plus y squared equals, say, 25. The graph of the 3 00:00:17,132 --> 00:00:23,024 points in the plane that satisfy this equation as a circle. But that's not the 4 00:00:23,036 --> 00:00:29,433 graph of a function, right? This graph fails the vertical line test. For a given 5 00:00:29,445 --> 00:00:35,337 input value, say 4 in this case, there's multiple y values which satisfy this 6 00:00:35,349 --> 00:00:41,727 equation. So, I can't simply solve this equation for y. Nevertheless, if you pick 7 00:00:41,739 --> 00:00:48,299 a specific point like 4, 3, you might be able to find a function whose graph traces 8 00:00:48,311 --> 00:00:54,350 out that same curve. So yeah, if I pick 4, 3, there is a function, y equals the 9 00:00:54,362 --> 00:01:00,800 square of 25 minus x squared. Which traces out a piece of the whole curve, right? I'm 10 00:01:00,812 --> 00:01:06,222 just ignoring the rest of this and this little tiny piece of the curve can be 11 00:01:06,234 --> 00:01:11,833 regarded as a function. If I had picked a different point, then I'm going to pick a 12 00:01:11,845 --> 00:01:16,906 different function. Instead of the square root of 25 minus x squared, if I wanted to 13 00:01:16,918 --> 00:01:21,870 stand down here, near the point 4, minus 3, well then maybe I'd pick the function y 14 00:01:21,882 --> 00:01:26,794 equals negative the square root of 25 minus x squared. If I ignore the rest of 15 00:01:26,806 --> 00:01:31,899 this and I'm just looking at this curve here, yet this curve by itself is a 16 00:01:31,911 --> 00:01:37,628 function. If I ignore this, it satisfies the vertical line test. This function is 17 00:01:37,640 --> 00:01:43,201 picking out a piece of the curve given by this equation which is, yeah, only valid 18 00:01:43,213 --> 00:01:48,600 near the point 4, minus 3. But maybe that's all I care about for the time 19 00:01:48,612 --> 00:01:52,850 being. So, let's say there is a function, y equals f of x, that satisfies the 20 00:01:52,862 --> 00:01:57,520 original equation. Well then, I can write that down. y equald f of x say satisfies 21 00:01:57,532 --> 00:02:00,925 the equation just means that x squared plus f of x squared equals 25. Now, I'm 22 00:02:00,937 --> 00:02:05,550 not saying that this gives me all of the solutions, right? The graph x squared plus 23 00:02:05,562 --> 00:02:09,940 y squared equals 25 is a circle fails the vertical line test. There is no function 24 00:02:09,952 --> 00:02:14,745 that gives me all those outputs because there's multiple outputs for a given 25 00:02:14,757 --> 00:02:20,119 input. All I'm saying is that I've got some function which traces out a piece of 26 00:02:20,131 --> 00:02:24,446 the whole curve. Then, I can differentiate. So, this is true for a 27 00:02:24,458 --> 00:02:29,457 bunch of values of x that I can differentiate this. The derivative of this 28 00:02:29,469 --> 00:02:34,796 sum is the sum of the derivative, so the derivative of x squared is 2x plus the 29 00:02:34,808 --> 00:02:40,058 derivative of f of x squared. I'm going to use a chain rule to do that. It's the 30 00:02:40,070 --> 00:02:46,731 derivative of the outside function at the inside times the derivative of the inside 31 00:02:46,743 --> 00:02:52,232 function equals the derivative of 25, which is zero. Now I can solve. So, 32 00:02:52,363 --> 00:02:57,927 subtract 2x from both sides and I'm left with 2 times f of x times f prime of x 33 00:02:57,939 --> 00:03:02,745 equals negative 2x. And then, I'll divide both sides by 2 times f of x. And I'll 34 00:03:02,757 --> 00:03:09,913 find that f prime of x is minus 2x over 2 f of x, and I can cancel those 2's and 35 00:03:09,925 --> 00:03:17,431 just get minus x over f of x. It seems like a funny situation. The derivative 36 00:03:17,443 --> 00:03:22,863 depends on more than just x. It also has an f of x. in it. 37 00:03:22,868 --> 00:03:27,804 Another way to say it is that the slope of the tangent line dy, dx, is negative x 38 00:03:27,816 --> 00:03:32,422 over y, right? y is f of x. And it does really seem a littie bit off putting 39 00:03:32,434 --> 00:03:37,600 initially in these kinds of calculations the slope of the tangent line depends on 40 00:03:37,612 --> 00:03:42,736 more than just x. It's negative x over y for this particular case. But think back 41 00:03:42,748 --> 00:03:47,440 to the piacture for this case, right? The picture's a circle. And what I'm 42 00:03:47,452 --> 00:03:51,595 saying is the slope of the tangent line is negative x over y. So, if you pick that 43 00:03:51,607 --> 00:03:56,295 point, say 4,3, and you ask what's the slope of the tangent line to the circle at 44 00:03:56,307 --> 00:04:00,815 the point 4,3 this equation is telling you the slope is -4 thirds. And yeah, that 45 00:04:00,827 --> 00:04:07,420 line is going down, the slope's negative. What's the slope of the tangent line to 46 00:04:07,432 --> 00:04:13,675 the curve at the point 4, negative 3? Same equation tells us that the slope there is 47 00:04:13,687 --> 00:04:19,110 4 3rds. And yeah, this line's going up. The slope of the tangent line is depending 48 00:04:19,122 --> 00:04:23,665 on more than just the x coordinate, right? You also need to know the y coordinate in 49 00:04:23,677 --> 00:04:28,275 order to know exactly what function you're actually looking at near that point. And 50 00:04:28,287 --> 00:04:32,566 that totally affects the slope of that tangent line. To do all these sorts of 51 00:04:32,578 --> 00:04:35,100 calculations, the trick is the cha in rule. 52 00:04:35,107 --> 00:04:39,591 For instance, if you're given some relation like this, x squared plus y cubed 53 00:04:39,603 --> 00:04:43,867 equals 1. You just got to make sure to think of y as a function of x. So that 54 00:04:43,879 --> 00:04:48,735 when you differentiate both sides, the derivative of the left hand side is 2x 55 00:04:48,747 --> 00:04:54,035 plus the derivative of y cubed equals the derivative of 1, which is 0, but what's 56 00:04:54,047 --> 00:04:59,285 the derivative of y cubed? If y is a function of x, then when you differentiate 57 00:04:59,297 --> 00:05:04,545 this, you've got to use the chain rule. It's 3 times the inside function squared, 58 00:05:04,652 --> 00:05:09,515 that's the derivative of the third power function, times the derivative of the 59 00:05:09,527 --> 00:05:14,655 inside function. I'll just write y prime. And as long as you're careful to use the 60 00:05:14,667 --> 00:05:19,250 chain rule, you'll be able to do these kinds of implicit differentiation 61 00:05:19,262 --> 00:05:24,180 problems. And you'll eventually solve for y prime in terms of both x and y. The 62 00:05:24,192 --> 00:05:25,705 chain rule is our friend.