When is a curve differentiable?
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- čas přidán 27. 06. 2024
- ► My Applications of Derivatives course: www.kristakingmath.com/applic...
0:00 // What is the definition of differentiability?
0:29 // Is a curve differentiable where it’s discontinuous?
1:31 // Differentiability implies continuity
2:12 // Continuity doesn’t necessarily imply differentiability
4:06 // Differentiability at a particular point or on a particular interval
4:50 // Open and closed intervals for differentiability
5:37 // Summary
When we talk about differentiability, it’s important to know that a function can be differentiable in general, differentiable over a particular interval, or differentiable at a specific point. In order for the function to be differentiable in general, it has to be differentiable at every single point in its domain. If there’s just a single point where the function isn’t differentiable, then we can’t call the entire curve differentiable. In that case, we could only say that the function is differentiable on intervals or at points that don’t include the points of non-differentiability.
So how do we determine if a function is differentiable at any particular point? Well, a function is only differentiable if it’s continuous. So if there’s a discontinuity at a point, the function by definition isn’t differentiable at that point. This applies to point discontinuities, jump discontinuities, and infinite/asymptotic discontinuities.
But there are also points where the function will be continuous, but still not differentiable. Remember, differentiability at a point means the derivative can be found there. If there derivative can’t be found, or if it’s undefined, then the function isn’t differentiable there.
So, for example, if the function has an infinitely steep slope at a particular point, and therefore a vertical tangent line there, then the derivative at that point is undefined. That means we can’t find the derivative, which means the function is not differentiable there. In the same way, we can’t find the derivative of a function at a corner or cusp in the graph, because the slope isn’t defined there, since the slope to the left of the point is different than the slope to the right of the point. Therefore, a function isn’t differentiable at a corner, either.
Therefore, in order for a function to be differentiable, it needs to be continuous, and it also needs to be free of vertical slopes and corners. Barring those problems, a function will be differentiable everywhere in its domain.
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Best calculus channel I ever watch.
Great explaination, this has made understanding the boundary conditions of schrodingers wave equation in quantum mechanics easier to digest.. thanks 😃👑
Ahhhh...why didn't my math professors teach like this!!!
Thank you for your videos, and for finding time to answer questions in the comments. Keep at it!
Thanks! I hope to. :)
You just made it so clear and obvious. Thanks a lot.
Thanks! I'm glad it could help. :)
Great quality as always :)
Thanks! :)
You are a lifesaver. Clear, and detailed explanation; like always, Thank you Krista King.
Thank you, Vincent! I'm so glad you liked it! :)
The music at the background for me😇 and bless your heart ❤️
Just subbed, I think you just might save my education lol thanks and make more!
p.s. where have you been my whole life!
Lol... I'm glad I can at least help now.
thank you for making these terms clear! I was confused when I first learned this material
Glad it could help! :)
So lucidly explained. Thank you a lot.
You're welcome, Rafat!
most helpful video on this topic on youtube!
BLESS YOUR SOUL !!!
Simple and clear explanation ! Good job :)
Thanks!
This is so awesome. Thank you.
:D
Always helpful!
This Great!; it's brief and concentrated and that make it very useful, I hope you create a video on integral like you did here to differentiation .. thanx for the good job
Thanks for the feedback! Yes, I'll be making one on integrals sometime as well. :)
You should do more style of videos like this in your future teaching videos! It's a great way of explaining the concept before applying it. I like it.
Thanks Kevin! Appreciate the feedback, and really glad you liked this! :D
Great explanation. Thank you very much!
You're welcome, Gabe! I'm so glad it helped! :)
idk when I subscribe to you krista but def. good video while im iceskating in NYC Happy Holidays to everyone and happy studying to all of ya.
your math is Kinging !!
Sounds like fun! :D Yes, have a great Christmas and Happy New Year!
Thank you! So helpful, especially with my calc final this Thursday :)
Good luck on your final! I hope you rock it. :)
Krista King Thanks! Hope I will too :D
Thank you! You helped so much!
You're welcome, Antonio, I'm so glad it helped! :D
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:D
Thanks for the video!
You bet, Andrea! :)
Fine explained
Excellent video lecture.
Thank you so much, Syamal! :)
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i love ur explanation
thx krista im from morroco and i like your vidoes
Hi! I'm glad you're liking them!
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Aw thanks! :)
Great Video Mrs. King, however I would have loved a shoutout to Weierstrass functions :)
I believe she has shouted it out though not mentioned the name. This makes the Weierstrass theorem easily understandable....me thinks!
great video
I like "continuous means you must be able to draw the graph without lifting your pencil."😃
thanks
:D
Gold explaining Gold
Hi, can u do a video on the difference between a function being differentiable on an 'open interval' and a function being differentiable on a 'closed interval'? Thanks.
yea
yea
Thank you so much
No problem, Layan! :)
thanks alot
oh my gosh thank you so much
You're so welcome, Aaron! :D
32. If the two functions f and g are differentiable at the number x1, is the composite function fºg necessarily differentiable at x1? If your answer is yes, prove it. If your answer is no, give a counterexample.
Could you please help me with this ex?? Thank so much.. Greetings from Perù!!
ok let me trying.. we know by chain`s rule Duy.Dxu when Dxu not problem.. The problem is with Duy, but if there is a guarantee for gx--->gx1 when x-->x1, you replace it and done.. but again, the example g(x)= sgnx , when g(x)=g(x1) dont let replace x1 for g(x1).. Is there a theorem I `m forgetting? sorry for my english..
Hi Krista, can you please tell me what software you use to make these videos? Thanks!
Hi, Kimberly! The software is called Sketchbook. It's made by Autodesk. :)
Can you please name the program of the chalkboard that you use in some of your videos? Thanks :)
I explain here: www.kristakingmath.com/blog/how-i-create-my-videos
Very nice
Thanks, Dinesh! :D
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Thanks!! :D
Good
How can a curve be differentiable on a closed interval? Wouldn't the fact that the end points are just points make it non-differentiable at those points? In this video there's an example of an open-interval differentiable function, but not a closed-interval one. In the example, the left-most point is included in the interval, but the function is still said to be non-differentiable at that point. I am confused by that, so any explanation would be helpful.
X^2+y^2=4 in circle can we say -2and+2 are points of differentiable, it means circle is differentiable
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Thanks!
Krista King can you please make a video on probability
Just a heads up, smooth usually means infinitely differentiable. So, smooth and continuous is redundant.
I think the graph in minute 3:00 is the cubic root, isn't it?
Yes it is!
wait wiat wait !!!
!!!.Any function differentiable then the function is continuous
can you explain to me how this rule goes with : 1\x or x^-1
its a differentiable function but its not continuous
#Thanks
It's not differentiable at x=0, even though it's differentiable everywhere else. :)
#Thanks_Krista
A blessing from the Lord!
Can a point have a slope?
If you really technically mean a single point, then no. But if you mean a particular point on a curve, then yes.
You are nice person
i hope i can meet you earler
y u upload after finals :(
Sorry about that! I wish I could have all of the videos done and published. I'm publishing when I can so that they will at least be out there when someone needs it in the future. :)
Good