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, Sometimes you don't have a function, you have a relation between two variables. A

classic example is x squared plus y squared equals, say, 25. The graph of the

points in the plane that satisfy this equation as a circle. But that's not the

graph of a function, right? This graph fails the vertical line test. For a given

input value, say 4 in this case, there's multiple y values which satisfy this

equation. So, I can't simply solve this equation for y. Nevertheless, if you pick

a specific point like 4, 3, you might be able to find a function whose graph traces

out that same curve. So yeah, if I pick 4, 3, there is a function, y equals the

square of 25 minus x squared. Which traces out a piece of the whole curve, right? I'm

just ignoring the rest of this and this little tiny piece of the curve can be

regarded as a function. If I had picked a different point, then I'm going to pick a

different function. Instead of the square root of 25 minus x squared, if I wanted to

stand down here, near the point 4, minus 3, well then maybe I'd pick the function y

equals negative the square root of 25 minus x squared. If I ignore the rest of

this and I'm just looking at this curve here, yet this curve by itself is a

function. If I ignore this, it satisfies the vertical line test. This function is

picking out a piece of the curve given by this equation which is, yeah, only valid

near the point 4, minus 3. But maybe that's all I care about for the time

being. So, let's say there is a function, y equals f of x, that satisfies the

original equation. Well then, I can write that down. y equald f of x say satisfies

the equation just means that x squared plus f of x squared equals 25. Now, I'm

not saying that this gives me all of the solutions, right? The graph x squared plus

y squared equals 25 is a circle fails the vertical line test. There is no function

that gives me all those outputs because there's multiple outputs for a given

input. All I'm saying is that I've got some function which traces out a piece of

the whole curve. Then, I can differentiate. So, this is true for a

bunch of values of x that I can differentiate this. The derivative of this

sum is the sum of the derivative, so the derivative of x squared is 2x plus the

derivative of f of x squared. I'm going to use a chain rule to do that. It's the

derivative of the outside function at the inside times the derivative of the inside

function equals the derivative of 25, which is zero. Now I can solve. So,

subtract 2x from both sides and I'm left with 2 times f of x times f prime of x

equals negative 2x. And then, I'll divide both sides by 2 times f of x. And I'll

find that f prime of x is minus 2x over 2 f of x, and I can cancel those 2's and

just get minus x over f of x. It seems like a funny situation. The derivative

depends on more than just x. It also has an f of x. in it.

Another way to say it is that the slope of the tangent line dy, dx, is negative x

over y, right? y is f of x. And it does really seem a littie bit off putting

initially in these kinds of calculations the slope of the tangent line depends on

more than just x. It's negative x over y for this particular case. But think back

to the piacture for this case, right? The picture's a circle. And what I'm

saying is the slope of the tangent line is negative x over y. So, if you pick that

point, say 4,3, and you ask what's the slope of the tangent line to the circle at

the point 4,3 this equation is telling you the slope is -4 thirds. And yeah, that

line is going down, the slope's negative. What's the slope of the tangent line to

the curve at the point 4, negative 3? Same equation tells us that the slope there is

4 3rds. And yeah, this line's going up. The slope of the tangent line is depending

on more than just the x coordinate, right? You also need to know the y coordinate in

order to know exactly what function you're actually looking at near that point. And

that totally affects the slope of that tangent line. To do all these sorts of

calculations, the trick is the cha in rule.

For instance, if you're given some relation like this, x squared plus y cubed

equals 1. You just got to make sure to think of y as a function of x. So that

when you differentiate both sides, the derivative of the left hand side is 2x

plus the derivative of y cubed equals the derivative of 1, which is 0, but what's

the derivative of y cubed? If y is a function of x, then when you differentiate

this, you've got to use the chain rule. It's 3 times the inside function squared,

that's the derivative of the third power function, times the derivative of the

inside function. I'll just write y prime. And as long as you're careful to use the

chain rule, you'll be able to do these kinds of implicit differentiation

problems. And you'll eventually solve for y prime in terms of both x and y. The

chain rule is our friend.