worthy hearth
Please help
#How do I prove these functions are not differentiable at point P?
delicate ibex
Since differentiation is concerned with limits, in layman’s terms, the gradient of the function when approaching from each side of the point are different.
E.g. in the first image you sent, when x -> 2-, the gradient is zero, but when x -> 2+, the gradient is 2. This means that then when x -> 2, there would be no way of evaluating the derivative at that point.
There’s probably a much more rigorous way of writing this out, but the overall gist is to do with when limits exist.
split shale
that is NOT in layman's terms lmao
delicate ibex
Hmm let's make it a little bit easier, sorry about that haha
Essentially, think about yourself travelling towards the point from the left hand side, and another friend travels from the right hand side.
From your perspective, you would say, coming from the negative side of P, that the gradient of the line would be 0.
From your friend's perspective, they would think the gradient at P, coming from the positive side, would be 2.
For the function to be differentiable at that point, both of you have to agree. Since you can't both agree on whether the gradient is 0 or 2, the derivative at that point doesn't exist
Hopefully an analogy is a lot more understandable, but as a high schooler, I sometimes struggle to explain things that well, since I've just borrowed things from online/textbooks lol
split shale
(also "gradient" is something different, maybe you should use the word "slope")
delicate ibex
Oh yh good point. At school, we use the words interchangeably, but there is probably some nuance to it
split shale
fair, but there is a nuance to it in multivariable calculus
delicate ibex
I see. Nice to learn something every day, so thanks!