Thread: Differentiable points

Hi, I'm having a hard time understanding how to determine where a function is not differentiable (the question keeps coming up in my homework though sadly, it didn't in the lesson!) I understand that a point where the function is not differentiable means the derivative does not exist there. However, how can I tell, based off either a graph or equation, when a point isn't differentiable?

For example, I'm currently doing this question:

Sketch the graph of f(x) = |x^2 - 1|

a) For what values of x is f not differentiable?
b) Find a formula for f1 and sketch the graph of f1
c) Find f1 at -2, 0, and 3.

Any help is greatly appreciated!

Originally Posted by starswept

Hi, I'm having a hard time understanding how to determine where a function is not differentiable

Use the limit definition of a derivative. if the limit is undefined at a certain point, then the function is not differentiable at that point. classic example, |x|. they tell you it's not differentiable at zero. why is that? simple, the limit does not exist at x = 0

you can use the apostrophe to denote a prime. don't say f1, say f '

Originally Posted by Jhevon

Use the limit definition of a derivative. if the limit is undefined at a certain point, then the function is not differentiable at that point. classic example, |x|. they tell you it's not differentiable at zero. why is that? simple, the limit does not exist at x = 0

you can use the apostrophe to denote a prime. don't say f1, say f '

So, essentially, saying the derivative doesn't exist is exactly like saying the limit does not exist? Does that mean that the right hand limit must equal the left hand limit rule still comes into play?

Originally Posted by starswept

So, essentially, saying the derivative doesn't exist is exactly like saying the limit does not exist? Does that mean that the right hand limit must equal the left hand limit rule still comes into play?

yes, all that comes into play. and also, if the limit diverges, that is, goes to infinity or something

Originally Posted by starswept

Hi, I'm having a hard time understanding how to determine where a function is not differentiable

Basically there are two conditions that one looks for in deciding if a function is differentiable.
The first is that the function must be continuous at the point!
The second is not that easily stated. But is essence it is that the graph of the function must be smooth at the point. If you graph f(x)=|x-1|, then even though it is continuous at x=1, you will see a sharp turn at x=1. Therefore, the function is not differentiable at x=1: it cannot have a tangent there.

Originally Posted by starswept

Hi, I'm having a hard time understanding how to determine where a function is not differentiable (the question keeps coming up in my homework though sadly, it didn't in the lesson!) I understand that a point where the function is not differentiable means the derivative does not exist there. However, how can I tell, based off either a graph or equation, when a point isn't differentiable?

For example, I'm currently doing this question:

Sketch the graph of f(x) = |x^2 - 1|

a) For what values of x is f not differentiable?
b) Find a formula for f1 and sketch the graph of f1
c) Find f1 at -2, 0, and 3.

Any help is greatly appreciated!

f(x)=|x-1||x+1| is not differentiable at x=-1 and x=1

Originally Posted by Plato

The second is not that easily stated. But is essence it is that the graph of the function must be smooth at the point. If you graph f(x)=|x-1|.

Which means the right and left derivatives are the same.