5:32 here I didn’t really understand why we don’t put the equal, Is it something that I have to do every time or there is a specific thing that tells me if I need to?
We need to investigate first before we can include the equal to. If g(x) is not differentiable at x = 2, we cannot include x = 2 in the domain of g'(x). If g(x) is differentiable at x = 2, then we can include x = 2 in the domain of g'(x).
i have a question isn’t it better to say derivative from the left and derivative from the right (LHD=RHD) and not limit of derivative, because the limit of derivative doesn’t have to exist for a function to be differentiable at a certain point the left and right hand derivatives must be equal not necessarily the limits of derivatives
Great explanation ! I have one question : In the last example , the reason for failing the differentiability test , is it because the limit does not prove that it is differentiable but if it were it would be equal to 1/2 ? If you use the epsilon-delta definition of the limit for the derivative , can you skip the continuity test ?
For a function to be differentiable, it must be continuous. We can have a situation where we have a function with a jump discontinuity, and the slopes to the right before and after the jump are equal (this is a differentiable trap). But that function would not be differentiable, because we need to be able to follow the slope of the tangent line immediately to the left and right to the point of tangency.
Thank you, it's helpful. But I have a question, what if you do the continuous part and you get the same values for the right and left side then for the differentiable part you find different values... For example maybe you are given a f(x) = 4 if x
The function you described is continuous but not differentiable at x = 2. For a function to be continuous, left/right check would have to work out for f and f'.
I'm sorry to bump like this... but the way you try to prove that g is differentiable at x=2 at 5:50 is wrong. It assumes that the derivative is continuous. There are certainly derivatives that turn out not to be continuous, so the logic is faulty. I agree that most exam-like examples will succeed in checking the lateral limits of the derivative being equal, corresponding to the actual lateral limits of the increment quotient... but still, it's not the way to prove differentiability.
I appreciate the comment! Thanks for doing a deep dive into the concept. The method I showed was for determining if a piecewise function was differentiable at a point- specifically for courses like AP Calculus or university calculus. I agree that we cannot assume the derivative is continuous if we want to write a rigorous proof. For a question that requires a formal proof/explanation, I would use the limit definition of differentiable. For a real analysis level course, we would have to go down the epsilon delta rabbit hole 🕳. Thanks for adding so much to the comment section!
I have a doubt "differentiable then it is continuous" But why you are solving first continuous then differentiable any other specific reason for it can you explain me as soon as possible
If a function is differentiable then it is continuous is a theorem in many calculus textbooks. I solve for continuity first because a function cannot be differentiable at a point if it is not connected at that point. The most common trap with differentiability is to find a piecewise function that has a jump discontinuity. Imagine a piecewise function that is made of two linear segments with equal slopes, but a jump discontinuity at x = 0. Even though the slope of the lines near 0 are the same, the function would not be differentiable at x = 0.
Good explanation. Straight, comprehensive, no jargon
I try to waste no time in these instructional videos, I am glad you appreciate that!
You just earned a subscriber because you reply so well to the comments
@Myheart19608 thanks for subscribing! Are you taking University Calc or AP Calc?
@@vinteachesmathyes,I am taking calculus classes
@@vinteachesmath uni calc
@@vinteachesmath University calc
straight to the point. That's what we need ! Thank you, man.
Glad this video was helpful!
5:32 here I didn’t really understand why we don’t put the equal, Is it something that I have to do every time or there is a specific thing that tells me if I need to?
We need to investigate first before we can include the equal to.
If g(x) is not differentiable at x = 2, we cannot include x = 2 in the domain of g'(x).
If g(x) is differentiable at x = 2, then we can include x = 2 in the domain of g'(x).
Thank you very much you made it easy for me. I really undertood. Thanks.
Glad it helped!
Thank you so much! Your explanations were easy to comprehend and you got straight to the point! Wish I had you as my teacher!
Thanks for supporting the channel! I try to get right to the point in my videos.
Thank you sir you made it much more easy😊
I am happy to help. I hope the rest of your school year goes well and thank you for supporting the channel!
Very helpful! Thank you
Glad it was helpful! I hope calculus goes well and thanks for supporting the channel!
Well explained!!!
I hope calculus is going well!
Thanks Man.
I appreciate it.
You're welcome! I hope your calculus class is going well.
Well Explained! Very Helpful.
i have a question isn’t it better to say derivative from the left and derivative from the right
(LHD=RHD) and not limit of derivative, because the limit of derivative doesn’t have to exist for a function to be differentiable at a certain point the left and right hand derivatives must be equal not necessarily the limits of derivatives
شكرآ ❤❤
Thank you 😊
Welcome! I hope calculus is going well.
Great explanation ! I have one question : In the last example , the reason for failing the differentiability test , is it because the limit does not prove that it is differentiable but if it were it would be equal to 1/2 ? If you use the epsilon-delta definition of the limit for the derivative , can you skip the continuity test ?
Thank you Sir
Most welcome!
thank you very much
Welcome! I hope your calculus class is going well.
If you want to check continuity and differentiability just see this video.
Straight forward❤❤❤
I appreciate the comment! I try to make these instructional videos as straightforward as possible. I hope you are doing well in calculus!
@@vinteachesmath❤❤
Hello , why do we have to show that g(x) is continuous first
For a function to be differentiable, it must be continuous. We can have a situation where we have a function with a jump discontinuity, and the slopes to the right before and after the jump are equal (this is a differentiable trap). But that function would not be differentiable, because we need to be able to follow the slope of the tangent line immediately to the left and right to the point of tangency.
Thank you sir
Thanks mannn
Happy to help! I hope calculus is going well!
Thank you, it's helpful.
But I have a question, what if you do the continuous part and you get the same values for the right and left side then for the differentiable part you find different values...
For example maybe you are given a f(x) = 4 if x
The function you described is continuous but not differentiable at x = 2. For a function to be continuous, left/right check would have to work out for f and f'.
Best
Thanks!
I'm sorry to bump like this... but the way you try to prove that g is differentiable at x=2 at 5:50 is wrong. It assumes that the derivative is continuous. There are certainly derivatives that turn out not to be continuous, so the logic is faulty. I agree that most exam-like examples will succeed in checking the lateral limits of the derivative being equal, corresponding to the actual lateral limits of the increment quotient... but still, it's not the way to prove differentiability.
I appreciate the comment! Thanks for doing a deep dive into the concept. The method I showed was for determining if a piecewise function was differentiable at a point- specifically for courses like AP Calculus or university calculus.
I agree that we cannot assume the derivative is continuous if we want to write a rigorous proof.
For a question that requires a formal proof/explanation, I would use the limit definition of differentiable. For a real analysis level course, we would have to go down the epsilon delta rabbit hole 🕳.
Thanks for adding so much to the comment section!
@@vinteachesmath
Did you assume anything ?
I have a doubt "differentiable then it is continuous" But why you are solving first continuous then differentiable any other specific reason for it can you explain me as soon as possible
If a function is differentiable then it is continuous is a theorem in many calculus textbooks. I solve for continuity first because a function cannot be differentiable at a point if it is not connected at that point.
The most common trap with differentiability is to find a piecewise function that has a jump discontinuity. Imagine a piecewise function that is made of two linear segments with equal slopes, but a jump discontinuity at x = 0. Even though the slope of the lines near 0 are the same, the function would not be differentiable at x = 0.
noice
Glad you think so! I hope you are doing well in calculus!