How to find whether this function is differentiable

In summary, the conversation is about determining the differentiability of the function F(x) at x=2 using the definition of derivative. The person trying to solve the problem initially gets a 0/0 limit for both sides, but after some algebraic manipulation, they are able to get both limits to equal 1/2, indicating that the function is differentiable at x=2.
  • #1
lax1113
179
0

Homework Statement


F(x)= {1/2x+1 when x=<2
{squareroot(2x) when x=>/=2

is it differentiable at x=2.


Homework Equations


(f(x)-f(2))/(x-2)


The Attempt at a Solution


So i know i ahve to take the limit from both the negative and positive of 2, and determine if they are equal. But after pluggin everyting in i get 0/0 for both equations, and i tried to change the equations a little bit to see if they could reveal the limit, but i coudln't figure it out.
 
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  • #2


You ALWAYS get a 0/0 limit when you apply the definition of derivative. You have to do some algebra to fix that. What form did you get for one of the limits and how did you try to 'fix' it so you get a finite limit?
 
  • #3


Hey dick,
Thanks for posting. I went back again a little bit after and it seemed really obvious. I hate how sometimes ill stare at a problem that is very simple and just don't get it, but then i revisit for 2 seconds and it jumps out at me.

I got them to each equal 1/2 after changing them algebraically.
 

FAQ: How to find whether this function is differentiable

1. How do I determine if a function is differentiable at a given point?

To determine if a function is differentiable at a point, you can use the definition of differentiability. This states that a function f(x) is differentiable at a point a if the limit of (f(x) - f(a))/(x - a) exists as x approaches a. If the limit exists, then the function is differentiable at that point.

2. Can a function be differentiable at some points but not others?

Yes, a function can be differentiable at some points but not others. This is because differentiability depends on the behavior of the function at a specific point. A function may be continuous at one point but not differentiable, or it may be differentiable at some points but not others.

3. What are the key properties of a differentiable function?

A differentiable function is continuous at the point of differentiability and has a well-defined derivative at that point. It also has a unique tangent line at that point and can be approximated by a linear function (its tangent line) near that point.

4. Can a function be differentiable but not continuous?

No, a function cannot be differentiable but not continuous. Differentiability implies continuity, as a function must be continuous at the point of differentiability in order for the limit to exist and the derivative to be defined.

5. How can I prove that a function is differentiable at a given point?

To prove that a function is differentiable at a point, you can use the definition of differentiability and show that the limit of (f(x) - f(a))/(x - a) exists as x approaches a. You can also use the differentiability rules, such as the sum, difference, product, and quotient rules, to determine if a function is differentiable at a point.

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