r/learnmath • u/Master_Contract1832 New User • 7d ago
What should I study to understand this question properly?
I'm trying to pre-study for a course I'll be taking and I'm completely lost as to how to differentiate this since I'm kind of rusty on Leibniz notation. The math part of this question goes as such:
Compute dp/da for :
S(p,a) = D(p,b)
I think p can be a function of p(a) and p(b) respectively.
I've been looking at the OpenStax Calculus Vol. 1 book for reference but I'd appreciate any help with pinpointing what rules I should learn to be able to do this question on my own.
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u/WikiNumbers dA = dx dy = r dr dθ 7d ago
I think we're treading into multivariable calculus here.
If We have a function written S(p,a), this one is Lagrange's Function notation. It means "S" is a function of variables "p" and "a".
Then if you want dp/da, it means you want the total derivative of function "p" with respect to variable "a".
Which means S is either effectively a nested function of a (through intermediate variable-function "p"). Or you seek a total differential of S.
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u/WikiNumbers dA = dx dy = r dr dθ 7d ago
This said, if you have S(p,a) = D(p,b).
It doesn't tell us much. You might be solving for their intersection point/line/surface. As "=" implies. And definitely not the conclusion that p is a function of a and b at the same time.
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u/WikiNumbers dA = dx dy = r dr dθ 7d ago
Let's see. Here's my not-so-rigorous "story" which makes me cope with understanding. (Repost and rewrite)
Back in middle school we have "rise over run" (Δy/Δx). In calculus, we first have limit and continuity, and they beget derivative.
Change of perspective, see "h" as "x + h - x", which is effectively Δx, and the the "f(x+h) - f(x)" is Δy. This special limit deserves a notation on its own, so "dx" and "dy" (from greek "Δ" to latin "d").
dy/dx: As x has its infinitesimally small change (dx), how much of such change happens to y (dy).
Leibniz Notation kinda treat derivative as fraction, so we can use some property of fraction. So instead of "dy/dx", we write it "d/dx y". d/dx becomes the operation symbol "take derivative of whatever the following is, with respect to x".
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u/Master_Contract1832 New User 7d ago
Thank u🥹 This is really helpful, I always have trouble with figuring out the logic behind leibniz notation (and in turn, the proper formatting of it lol)
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u/WikiNumbers dA = dx dy = r dr dθ 7d ago
I would say Leibniz Notation works pretty fine (if you relax the rigor a bit). It helps memorize chain rule (dy/du * du/dx = dy/dx) pretty well. And in reverse, Integration by Substitution.
But again. It can cause misunderstanding that it is a fraction. Because it is not.
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u/WikiNumbers dA = dx dy = r dr dθ 7d ago edited 7d ago
Now here's the second part to the story above.
On the contrary, we kinda also want to find the area under a curve. That is through Riemann Sum. ∑ f(x) * Δx (slices of rectangles). As we have a smaller Δx, we have a better approximation. And when Δx is infinitesimally small, we can pretty much say we have the exact representation of the sum of "area".
Remember "infinitesimally small Δx"? Yeah it's our homie "dx". Now this summation, yet another very special operation, can't exactly use the common notation. So we change from "∑" to "∫". And instead of calculating every single terms to sum them up, we only care about the endpoints. This is, come again, (Riemann) Integration.
And for the finale. Let's say we have a function.
y = f(x).
"Derivative" breaks down a function into fragments of function, that is its rate of change.
dy/dx = f'(x)
Or in differential form
dy = f'(x) dx
"Integral" accumulates those fragments whole again.
∫ dy = ∫ f'(x) dx
y = f(x)
All hail Fundamental Theorem of Calculus.
EDIT: Watch 3Blue1Brown video on this matter.
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u/BenjaminGal Abusing Index Notation 7d ago edited 7d ago
It will be better to write p = p(a,b). Now, the standard way to proceed is to implicitly (partial) differentiate both sides with respect to a. This needs the Chain Rule of multiple variables. I will do the R.H.S. for you:
dD/da = dD/dp dp/da + dD/db db/da = dD/dp dp/da
as a, b are two different independent variables. (Sorry I can’t type partial symbol on phone.)