Prove relationship between right derivative and derivative: f'+(a) = lim x->a+ f'(x)

[usk]

I don't know how to prove that:
f'+(a) = lim x->a+ f'(x)

 

[stapel]

I don't know how to prove that:
f'+(a) = lim x->a+ f'(x)

I'm sorry, but I don't understand what you're saying in the above. Are you given an expression for what [imath]f(x)[/imath] is? Why are you adding [imath]a[/imath] to [imath]f'[/imath]? Why does [imath]f[/imath]' have an argument (namely, [imath]x[/imath]) on the right-hand side of the equation, but not on the left-hand side? Is the limit on the right-hand side "as [imath]x[/imath] approaches [imath]a[/imath] from the right, so it's actually the following?

[imath]\qquad \displaystyle{ \lim_{x \rightarrow a^+} f'(x) }[/imath]

What do you mean by taking the limit of the derivative? Are you trying to prove something about second derivatives?

When you reply, please include a clear listing of your thoughts and efforts so far, so that we can see what's going on. ("Read Before Posting") Thank you!

 

[Dr.Peterson]

I don't know how to prove that:
f'+(a) = lim x->a+ f'(x)

I think what you mean to say may be this: [math]f'_+(a) =\lim_{x\to a^+} f'(x)[/math]
But I think this is almost the definition of the right derivative.

In any case, you will need to state the conditions for the claim. Is the function known to be differentiable somewhere, for example? And what have you learned that you can use to prove this?

 

[usk]

I think what you mean to say may be this: [math]f'_+(a) =\lim_{x\to a^+} f'(x)[/math]

Correct

But I think this is almost the definition of the right derivative.

"almost": that's the problem the right derivative is defined as:
[math]f'_+(x) =\lim_{h\to 0^+} \frac{f(x+h) - f(x)}{h}[/math]
How does it follow from the definition that
[math]f'_+(a) =\lim_{x\to a^+} f'(x)[/math]?

In any case, you will need to state the conditions for the claim. Is the function known to be differentiable somewhere, for example?

f: R -> R is differentiable on R

And what have you learned that you can use to prove this?

Given high school math:
- definition of limit
- limit laws (sum, product, quotient, exponent, constant product, constant)
- how to evaluate a limit (direct substitution, indeterminate forms, common limits like [math]\lim_{x\to 0} \frac{sin(x)}{x} = 1[/math] )
- the definition of derivative
- the definition of right derivative
Prove:
[math]f'_+(a) =\lim_{x\to a^+} f'(x)[/math]
I have no clue

 

[blamocur]

What about [imath]f(x) = x^2 \sin\frac{1}{x}[/imath], [imath]a=0[/imath], and [imath]f(0)=0[/imath] ? As far as I can tell, [imath]f^\prime_+(a)[/imath] exists, but [imath]\lim_{x\rightarrow 0^+} f^\prime(x)[/imath]does not.

 

[usk]

Suppose the limit exists

 

[blamocur]

Suppose the limit exists

What do you mean? Can you figure out [imath]f^\prime(x)[/imath] for [imath]x\neq 0[/imath] ?