difference quotient to find derivative of f(x) = 4

[PaulKraemer]

Hi,

I am having trouble using the diffence quotient method to find the derivative at x = 0 of f(x) = 4

So far I have tried the following...

f'(0) = lim as x->0 of (f(x) - f(0)) / (x - 0) = (4 - 4) / (0 - 0) = 0/0 ?????

I know that the derivative is zero...I just can't figure out how to show this using the difference quotient method. If anyone could help me out with this, I'd really appreciate it.

Thanks in advance,
Paul

 

[JeffM]

Hi,

I am having trouble using the diffence quotient method to find the derivative at x = 0 of f(x) = 4

So far I have tried the following...

f'(0) = lim as x->0 of (f(x) - f(0)) / (x - 0) = (4 - 4) / (0 - 0) = 0/0 ?????

I know that the derivative is zero...I just can't figure out how to show this using the difference quotient method. If anyone could help me out with this, I'd really appreciate it.

Thanks in advance,
Paul

You tend to forget what you learned about the limit of a function when you see the limit as h approaches 0 of [f(x + h) - f(x)] / h because you learned about the limit of f(x) when studying limits. BUT f(x) IS NOT THE FUNCTION WHOSE LIMIT YOU SEEK HERE.
Set, for all x except x = 0, g(x) = [f(x) - f(0)] / (x - 0) = (4 - 4) / x = 0.
What is the limit at 0 of g(x) when g(x) = 0 everywhere but x = 0?
You ignore what happens exactly at the limit, right?

 

[Deleted member 4993]

\(\displaystyle f'(x) \ = \ \lim_{h \to 0}\left [\frac{f(x+h) - f(x)}{h}\right ]\)

for the given function:

f(x) = 4

\(\displaystyle f'(x) \ = \ \lim_{h \to 0}\left [\frac{f(x+h) - f(x)}{h}\right ] \ = \ \lim_{h \to 0}\left [\frac{4 - 4}{h}\right ] \ = \ \lim_{h \to 0}\left [\frac{0}{h}\right ] \ = \ \lim_{h \to 0} [0] \ = 0\)

 

[lookagain]

I am having trouble using the diffence quotient method to find the derivative at x = 0 of f(x) = 4

Notice:

f(x) = 4 is a constant function. The graph is a horizontal line with a slope 0. Everwhere on it
the slope is 0. The derivative at x = 0, then, is the slope of the function there, which is 0.

Keep this mind for a check as you do the limit of the difference function for this function.