Is f(x) differentiable at x=1?

In summary, the conversation discusses how to determine if a function is differentiable at a specific point, using the example of a piece-wise function. The process involves finding the reasons why the limit may not exist, computing the derivative on either side of the specific point, and determining if the derivative is continuous. The conversation also mentions that while a derivative function is not necessarily continuous, it is continuous where defined. The conversation concludes by providing examples of finding derivatives and their values at a specific point.
  • #1
cyt91
53
0
Given
http://www.mathhelpforum.com/math-help/vlatex/pics/105_fde5ac6b051b4fac473487c7b4afa9e5.png

Is f(x) differentiable at x=1?

I know that we have to prove
http://www.mathhelpforum.com/math-help/vlatex/pics/65_6fae3c52eaa96aaafdf2c225a900ea48.png

exist/does not exist at x=1. But how do I begin? It's a piece-wise function.

Thanks for your help.
 
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  • #2
Start by finding out the reasons that limit might not exist. This is a common type of problem, so you should find examples in your textbook.
 
  • #3
The idea is one sided limits. You can compute the derivatives of f for x>1, and x<=1, for the derivative f' to exist, f' must be continuous there. So you've reduced the problem to continuity.
 
  • #4
To add to hunt mat's response, while a derivative function is not necessarily continuous, it is continuous where ever it is defined. That is why taking the left and right side limits of the derivative works- if they are the same, the function is differentiable and the joint value is the derivative at that point.

What is the derivative of [itex]x^2[/itex]? What is its value at x= 1? What is the derivative of (x+ 1)/2? what is its value at x= 1?
 

FAQ: Is f(x) differentiable at x=1?

1. What does it mean for a function to be differentiable at a specific point?

When a function is differentiable at a specific point, it means that the function is smooth and has a well-defined slope or derivative at that point. This means that the function has a tangent line that can be drawn at that point without any breaks or sharp turns.

2. How do you determine if a function is differentiable at a given point?

To determine if a function is differentiable at a given point, you can use the definition of differentiability, which states that a function is differentiable at a point if the limit of the difference quotient (or slope) exists as the change in x approaches 0. Alternatively, you can also use graphical methods, such as checking if the function has a smooth graph with no sharp turns or discontinuities at that point.

3. What is the difference between continuity and differentiability?

Continuity and differentiability are related concepts, but they are not the same. A function is continuous at a point if the limit of the function exists at that point. On the other hand, a function is differentiable at a point if the limit of the difference quotient (or slope) exists at that point. In other words, continuity is about the existence of a limit, while differentiability is about the existence of a derivative.

4. Can a function be differentiable at a point but not continuous?

No, a function cannot be differentiable at a point if it is not continuous at that point. This is because differentiability requires the limit of the difference quotient to exist, which can only happen if the function is continuous at that point. However, it is possible for a function to be continuous at a point but not differentiable, as differentiability also requires the function to be smooth and have a well-defined slope at that point.

5. What is the significance of a function being differentiable at a point?

A function being differentiable at a point has several important implications. First, it means that the function has a well-defined slope or derivative at that point, which can be useful for finding the rate of change of the function. Additionally, differentiable functions are also typically easier to work with mathematically, as they have well-defined tangent lines and can be approximated by linear functions. Finally, differentiability is a necessary condition for a function to be continuously differentiable and have a power series representation.

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