Help with the Chain Rule
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Help with the Chain Rule
I was recently taught about the Chain Rule in calculus class, and I am really confused. My biggest problem is:
(d/dx) f(x) = f'(x) And
(d/dx) f(u) = f'(u) * u' Substitute (d/dx) f(u) with f'(u) and you get
f'(u) = f'(u) * u'
While I am asking questions, how is a corner/ cusp different than a curve, and what is a tangent line. I was told that it is a line that touches a graph (locally) at only one point, but then a vertical line is the tangent of any point of a function by definition of a function.
(d/dx) f(x) = f'(x) And
(d/dx) f(u) = f'(u) * u' Substitute (d/dx) f(u) with f'(u) and you get
f'(u) = f'(u) * u'
While I am asking questions, how is a corner/ cusp different than a curve, and what is a tangent line. I was told that it is a line that touches a graph (locally) at only one point, but then a vertical line is the tangent of any point of a function by definition of a function.
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Re: Help with the Chain Rule
the second equation is (without using prime notation)
(d/dx) f(u) = (d/du) f(u) * (d/dx) u
and it is not the case that (d/dx) f(u) = (d/du) f(u) = f'(u) so the substitution does not work
intuitively, the tangent line lies "parallel" to the curve at the point it is touching, although wikipedia gives a more rigorous definition
(d/dx) f(u) = (d/du) f(u) * (d/dx) u
and it is not the case that (d/dx) f(u) = (d/du) f(u) = f'(u) so the substitution does not work
intuitively, the tangent line lies "parallel" to the curve at the point it is touching, although wikipedia gives a more rigorous definition
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Re: Help with the Chain Rule
For another intuition, a tangent touches a curve in exactly one point but does not intersect it. (At least locally  "far" from the tangent point, the line might intersect the curve, but we're typically just interested in the "local" behavior when talking about tangents.)
When talking about tangent lines in calculus, tho, that "it's parallel to the point it's touching" is the intuition you typically want  it's "what would happen if the curve just kept going straight at this point, rather than continuing to curve?".
When talking about tangent lines in calculus, tho, that "it's parallel to the point it's touching" is the intuition you typically want  it's "what would happen if the curve just kept going straight at this point, rather than continuing to curve?".
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Re: Help with the Chain Rule
Xanthir wrote:For another intuition, a tangent touches a curve in exactly one point but does not intersect it. (At least locally  "far" from the tangent point, the line might intersect the curve, but we're typically just interested in the "local" behavior when talking about tangents.)
Doesn't quite work for example with y = x^{3} at the point (0, 0), where the tangent line y = 0 does in fact intersect the curve at that point.
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Re: Help with the Chain Rule
Yeah, the "not intersecting" specification should not be considered part of the definition of tangency at all.
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Re: Help with the Chain Rule
It's not, it's part of an intuition about tangency, to distinguish the tangent from all the other lines that just intersect the curve at one point.
It's definitely not exact.
It's definitely not exact.
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Re: Help with the Chain Rule
A tangent line is simply the line that best approximates the curve when you zoom in reeeeally close to a given point. Lines are generally much easier to work with than curves, so looking at a tangent line instead can give you a good sense of what the curve is doing, when the curve itself may be too complicated to work with. This kind of thinking happens all the time in engineering.

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Re: Help with the Chain Rule
>) wrote:the second equation is (without using prime notation)
(d/dx) f(u) = (d/du) f(u) * (d/dx) u
and it is not the case that (d/dx) f(u) = (d/du) f(u) = f'(u) so the substitution does not work
You are taking the derivative of a function with respect to another function. What does that mean? Also, if we are allowed to take derivatives with respect to functions, then what does the derivative of a function with respect to itself equal?
P.S. I am fairly certain of this, but I just want to check; when we write u, we actually mean u(x).
Wikipedia wrote:More precisely, a straight line is said to be a tangent of a curve y = f (x) at a point x = c on the curve if the line passes through the point (c, f (c)) on the curve and has slope f '(c) where f ' is the derivative of f.
My teacher said that the only time a function has a limit, and is not differentiable is when there is a corner, cusp, or a vertical tangent. Therefor, Wikipedia's definition of a tangent is flawed.
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Re: Help with the Chain Rule
jewish_scientist wrote:You are taking the derivative of a function with respect to another function. What does that mean? Also, if we are allowed to take derivatives with respect to functions, then what does the derivative of a function with respect to itself equal?
P.S. I am fairly certain of this, but I just want to check; when we write u, we actually mean u(x).
The limit definition you learned stays good, as does the general intuition for da/db as the ratio of the change in a to the change in b when the change in b gets very small. And the ratio of the change in u to the change in u is always 1 (including when the change in u is small.)
My teacher said that the only time a function has a limit, and is not differentiable is when there is a corner, cusp, or a vertical tangent. Therefor, Wikipedia's definition of a tangent is flawed.
The functions you are dealing with so far are generally of the form y(x). There, there is something special going on with verticalness. Wikipedia is using a notion of tangency which is appropriate to math more advanced than calc 1. It just gets better and better! If you think of a function parametrized like (x(t),y(t)), it will be very comfortable indeed to talk about vertical lines as tangents.
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Re: Help with the Chain Rule
What's the volume of a sphere, as a function of its radius? V(r) = 4/3 pi r³, right? So what's the derivative of the volume with respect to the radius? That would be dV/dr = 4 pi r² (which happens to be the surface area, which is a nifty thing, but let's not dwell on that for now). So in other words, the rate at which the volume changes as the radius changes is proportional to the square of the radius itself. With me so far?
Well now, what if the radius is, itself, a function of something else? Maybe we're inflating a spherical balloon such that r(t) = sqrt(t), i.e. it increases as the square root of time. So at t = 0, r = 0, and at t = 4, r = 2, and so forth. Then dr/dt = 1/2 t^{1/2}, I think you'll find. Well so what? Well, how is the volume of the balloon changing as a result?
We can just substitute directly in this case, giving us V(t) = V(r(t)) = 4/3 pi t^{3/2} and then differentiating to get V'(t) = 2 pi t^{1/2}. Or, we can use the chain rule. Knowing that dV/dt = dV/dr × dr/dt, we get dV/dt = 4 pi r² × 1/2 t^{1/2} = 4 pi t × 1/2 t^{1/2} = 2 pi t^{1/2}.
Sometimes, substitution isn't quite so pretty, and the chain rule is the only way to make it work. And you can even use it to pull tricks like 1 = dy/dy = dy/dx × dx/dy, hence the derivative of y with respect to x is the inverse of the derivative of x with respect to y, i.e. at the equivalent points of inverse functions, the slopes of the tangents are inverses of each other.
As for more complicated stuff, check what doogly just said. Once you start introducing extra dimensions everything gets fun and you can do stuff you never thought possible.
Well now, what if the radius is, itself, a function of something else? Maybe we're inflating a spherical balloon such that r(t) = sqrt(t), i.e. it increases as the square root of time. So at t = 0, r = 0, and at t = 4, r = 2, and so forth. Then dr/dt = 1/2 t^{1/2}, I think you'll find. Well so what? Well, how is the volume of the balloon changing as a result?
We can just substitute directly in this case, giving us V(t) = V(r(t)) = 4/3 pi t^{3/2} and then differentiating to get V'(t) = 2 pi t^{1/2}. Or, we can use the chain rule. Knowing that dV/dt = dV/dr × dr/dt, we get dV/dt = 4 pi r² × 1/2 t^{1/2} = 4 pi t × 1/2 t^{1/2} = 2 pi t^{1/2}.
Sometimes, substitution isn't quite so pretty, and the chain rule is the only way to make it work. And you can even use it to pull tricks like 1 = dy/dy = dy/dx × dx/dy, hence the derivative of y with respect to x is the inverse of the derivative of x with respect to y, i.e. at the equivalent points of inverse functions, the slopes of the tangents are inverses of each other.
As for more complicated stuff, check what doogly just said. Once you start introducing extra dimensions everything gets fun and you can do stuff you never thought possible.
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Re: Help with the Chain Rule
One day, the ultimate power of the tangent bundle may be yours!
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Re: Help with the Chain Rule
The chain rule applies for nested functions. I like to think of them as having an inner and out part. I teach my students, in addition to the notation you've written that the chain rule is the following sentence:
"the derivative of the outside, with the inside unchanged, times the derivative of the inside."
For some (most so far, but it's only my 3rd year teaching Calc) that is a helpful way to see the chain rule.
Are you learning this at college or high school? I'm asking b/c if it's high school that makes me feel good about the pacing of my class!
"the derivative of the outside, with the inside unchanged, times the derivative of the inside."
For some (most so far, but it's only my 3rd year teaching Calc) that is a helpful way to see the chain rule.
Are you learning this at college or high school? I'm asking b/c if it's high school that makes me feel good about the pacing of my class!

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Re: Help with the Chain Rule
Repeekthgil wrote:The chain rule applies for nested functions. I like to think of them as having an inner and out part. I teach my students, in addition to the notation you've written that the chain rule is the following sentence:
"the derivative of the outside, with the inside unchanged, times the derivative of the inside."
For some (most so far, but it's only my 3rd year teaching Calc) that is a helpful way to see the chain rule.
Yeah, that is how my teacher describes it. One part that I have trouble wrapping my head around is that it does not matter where the inner function is.
f(u) = stuff + u then d/dx(u) = d/du f(u) * d/dx u(x)
f(u) = stuff * u then d/dx(u) = d/du f(u) * d/dx u(x)
f(u) = stuff / u then d/dx(u) = d/du f(u) * d/dx u(x)
Are you learning this at college or high school? I'm asking b/c if it's high school that makes me feel good about the pacing of my class!
My high school (I really did not like that place) required 3 years of math. However, one of those unwritten rules of what was that all seniors take calculus or statistics. I decided to do neither. The only classes I took were studies, sciences and 1 English (it was required). I think I had 4 sciences that year. They were physics, disease diagnostics (a.k.a. House the class),Destination Imagination, and engineering. It was my best school year, so far. So I am in college now, I did have the opportunity to take calculus.
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Re: Help with the Chain Rule
jewish_scientist wrote:Yeah, that is how my teacher describes it. One part that I have trouble wrapping my head around is that it does not matter where the inner function is.
f(u) = stuff + u then d/dx(u) = d/du f(u) * d/dx u(x)
f(u) = stuff * u then d/dx(u) = d/du f(u) * d/dx u(x)
f(u) = stuff / u then d/dx(u) = d/du f(u) * d/dx u(x)
Actually none of these seem right. By d/dx(u), do you mean du/dx? That wouldn't involve any chain rule. If you mean something else, the notation is very weird.
If you have f(u), df/dx = df/du * du/dx.
And then it ought to make sense that the chain rule doesn't depend on what f(u) is at all. Right? What the chain rule is saying is that
df/dx = df/du * du/dx
How f(u) looks will matter an awful lot for what df/du is. If you go to actually implement these things, it makes a big difference.
But the chain rule itself is about partitioning these dependencies. It doesn't care about the actual functional form of f(u), or of u(x).
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Re: Help with the Chain Rule
doogly wrote:jewish_scientist wrote:Yeah, that is how my teacher describes it. One part that I have trouble wrapping my head around is that it does not matter where the inner function is.
f(u) = stuff + u then d/dx(u) = d/du f(u) * d/dx u(x)
f(u) = stuff * u then d/dx(u) = d/du f(u) * d/dx u(x)
f(u) = stuff / u then d/dx(u) = d/du f(u) * d/dx u(x)
Actually none of these seem right. By d/dx(u), do you mean du/dx? That wouldn't involve any chain rule. If you mean something else, the notation is very weird.
I meant d/dx f(u). I messed up on the first, and then copied it for the others.
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Re: Help with the Chain Rule
Word. Does the explanation of why make sense?
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Re: Help with the Chain Rule
Yeah, I guess so. I bet it will make more sense when I solve more problems and see how it hows for equations that are structured differently. Still, I do not understand how is a corner/ cusp different than a curve, and what is a tangent line.
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Re: Help with the Chain Rule
A corner / cusp is a feature of a curve. You would say things like, "This curve has a cusp here," or, "This curve is continuous but not smooth, look, it has a 90 degree corner right there."
A tangent to a curve f at a point p is one which coincides with the curve at that point, and has the same slope as the curve / points in the same direction as the curve does at that point.
In the calculus sense, you can imagine the tangent as the limit of secant lines. If you took two points on a curve, f(p) and f(q), you could draw a line connecting them (the name of that line is the secant.) If you slide q towards p, and by slide, I mean take the limit as q approaches p, your secant lines approach what will be called the tangent line.
You took physics before this, right? If you use your physical intuition for instantaneous velocity as the mathematical benchmark for tangent, you are in the right place. (Though the magnitude of velocity means something that doesn't correspond generally. For tangents, we will either want to draw an infinite line or talk about a unit vector.)
A tangent to a curve f at a point p is one which coincides with the curve at that point, and has the same slope as the curve / points in the same direction as the curve does at that point.
In the calculus sense, you can imagine the tangent as the limit of secant lines. If you took two points on a curve, f(p) and f(q), you could draw a line connecting them (the name of that line is the secant.) If you slide q towards p, and by slide, I mean take the limit as q approaches p, your secant lines approach what will be called the tangent line.
You took physics before this, right? If you use your physical intuition for instantaneous velocity as the mathematical benchmark for tangent, you are in the right place. (Though the magnitude of velocity means something that doesn't correspond generally. For tangents, we will either want to draw an infinite line or talk about a unit vector.)
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Re: Help with the Chain Rule
I understand what a corner, cups, and tangent are; what I want to know is what their mathematical definitions are.
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Re: Help with the Chain Rule
Corner, I don't know? I think it's where you just have a piecewise definition of the function.
Cusps are point in a curve which are continuous, and a derivative from the left and the right exist, but they don't agree. Like, if you looked at a { shape, the point at the tip would have two different tangents you could define, whether you are coming from the top or the bottom.
A tangent to a curve f at a point p is one which passes through f(p) and has slope equal to the derivative. You can use y = mx+ b with m = df/dx evaluated at p if you are in calc 1. If you want to worry about vertical lines, then you need to use the parametric form, and then the derivative will be well defined and you can draw your vertical line.
Cusps are point in a curve which are continuous, and a derivative from the left and the right exist, but they don't agree. Like, if you looked at a { shape, the point at the tip would have two different tangents you could define, whether you are coming from the top or the bottom.
A tangent to a curve f at a point p is one which passes through f(p) and has slope equal to the derivative. You can use y = mx+ b with m = df/dx evaluated at p if you are in calc 1. If you want to worry about vertical lines, then you need to use the parametric form, and then the derivative will be well defined and you can draw your vertical line.
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Re: Help with the Chain Rule
I think doogly may be mixing up the terms a little bit. Or at least, it is different than what I was taught.
To understand the mathematical definitions, you need to be pretty familiar with the concept of limits, which can get surprisingly technical, but hopefully makes some intuitive sense. The derivative of the function f(x) at point a is the limit as x approaches a of the ratio (f(x)f(a))/(xa). If you think you understand what a tangent line is, the derivative is just the slope of the tangent line at any point along the curve. No big deal. In fact, this is basically the mathematical definition of a tangent line of the curve f(x) at point a: it's the line that passes through the point (a,f(a)) that has the same derivative as f(x) at point a.
Now that the TLDR of differential calculus is out of the way, you can define left and right derivatives by choosing values of x that are only to the left or right of the value a, respectively. A corner is where the left and right derivatives exist (or one is infinite), but they aren't equal to each other. For example, the absolute value function has a corner at x=0 because it has derivative +1 to the right of the origin, and 1 to the left. Since the curve looks like the line y=x for x>0 and y=x for x<0, you can't really assign a single tangent line at x=0.
A continuous function has a cusp at point a if its left and right derivatives are both infinite, but they have different signs. Let's look at the curve y=x^(2/5). The curve becomes more and more vertical near the origin (the derivative approaches +/ infinity as x approaches 0). However, the derivative approaches +infinity from the right and infinity from the left. Hence the cusp.
A definition I find more helpful comes from a bug crawling along the curve at constant speed. If the bug's direction always changes smoothly and gradually, then there is a tangent line at every point in the curve. The bug passes through a cusp if it must make a sudden 180° turn, and it passes through a corner if it has to turn by any other angle.
To understand the mathematical definitions, you need to be pretty familiar with the concept of limits, which can get surprisingly technical, but hopefully makes some intuitive sense. The derivative of the function f(x) at point a is the limit as x approaches a of the ratio (f(x)f(a))/(xa). If you think you understand what a tangent line is, the derivative is just the slope of the tangent line at any point along the curve. No big deal. In fact, this is basically the mathematical definition of a tangent line of the curve f(x) at point a: it's the line that passes through the point (a,f(a)) that has the same derivative as f(x) at point a.
Now that the TLDR of differential calculus is out of the way, you can define left and right derivatives by choosing values of x that are only to the left or right of the value a, respectively. A corner is where the left and right derivatives exist (or one is infinite), but they aren't equal to each other. For example, the absolute value function has a corner at x=0 because it has derivative +1 to the right of the origin, and 1 to the left. Since the curve looks like the line y=x for x>0 and y=x for x<0, you can't really assign a single tangent line at x=0.
A continuous function has a cusp at point a if its left and right derivatives are both infinite, but they have different signs. Let's look at the curve y=x^(2/5). The curve becomes more and more vertical near the origin (the derivative approaches +/ infinity as x approaches 0). However, the derivative approaches +infinity from the right and infinity from the left. Hence the cusp.
A definition I find more helpful comes from a bug crawling along the curve at constant speed. If the bug's direction always changes smoothly and gradually, then there is a tangent line at every point in the curve. The bug passes through a cusp if it must make a sudden 180° turn, and it passes through a corner if it has to turn by any other angle.
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Re: Help with the Chain Rule
Word, those are the right words.
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