Determining if a function is differentiable at the indicated point

In summary, for a function to be differentiable at a point, its left and right derivatives must be equal. This function has two different pieces, with different derivatives on the left and right of x=1. To evaluate if F'(x) is an integer with no variable, simply plug in the given value of x and solve for F'(x). If there is no variable, the function is constant and the derivative will be a constant value.
  • #1
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How do you determine this?

F(x) x^2 +1 if x<1
F(x) 2x if x >= 1 at x=1

Are there designated steps? I understand that it is the derivative, but I don't understand the differentiable at the indicated point part..
 
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  • #2
For a function to be differentiable at a point, its left sided and right sided derivatives must be equal. Usually the function is the same on both sides of a point, but for this function it has two different pieces.
 
  • #3
I don't understand how is it different on the left sided and right sided? Would you put 1 for the value of x and get a solution that way?
 
  • #4
On the "left" of 1, the function is defined as F(x)=x^2 +1, then F'(1)=_____
On the "right" of 1, the function is defined as F(x)=2x, then F'(1)=_____

If these two values exist and are equal, then the function is differentiable at 1.
 
  • #5
Ok so only if first there is a vale for f'(x) and if the two valeus match is the function "differentiable" at the indicated point...Awesome! THANK YOU
One more question if you would be so kind...how do you evaluate if F'(x) = an integer with no variable? F'(1) = 2 since F(x) =2x
 
  • #6
If there's no variable then the function doesnt...vary. ie no matter what x is, the function is just constant.
 

1. What is the definition of differentiability at a point?

Differentiability at a point means that the function is continuous at that point and has a well-defined slope or derivative at that point.

2. How can I determine if a function is differentiable at a point?

To determine differentiability at a point, you can use the definition of differentiability or apply the rules for differentiating functions to see if the limit of the difference quotient exists at that point.

3. What are the key conditions for a function to be differentiable at a point?

The key conditions for a function to be differentiable at a point are that the function must be continuous at that point, and the limit of the difference quotient must exist at that point.

4. Can a function be differentiable at some points and not others?

Yes, a function can be differentiable at some points and not others. This usually occurs when the function has a sharp corner or a vertical tangent at a specific point, making the limit of the difference quotient undefined at that point.

5. How does the graph of a differentiable function look like at a point?

If a function is differentiable at a point, the graph of the function will have a smooth and continuous tangent line at that point. This means that the function will have a well-defined slope or derivative at that point.

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