# Domain of a Derivative

#### purplec16

I have a quick question: what is domain of the derivative of f(x) = 5?

#### Ackbeet

MHF Hall of Honor
I have two questions for you:

1. What is the domain of f(x)?
2. What is the derivative of f(x)?

#### purplec16

The domain of f(x) is all real numbers

the derivative of f(x) is:

$$\displaystyle \lim_{h \to \0}\frac{f(x+h)-f(x)}{h }$$

$$\displaystyle \lim_{h \to \0}\frac{5-5}{h }$$

$$\displaystyle \lim_{h \to \0}\frac{0}{h }$$

= $$\displaystyle \frac{0}{0 }$$

#### Ackbeet

MHF Hall of Honor
I would agree with all but your last line of the computation. h never actually equals zero (it's a limit!), whereas the numerator is identically zero. Hence the overall limit is what?

I was asking about the domain of f(x), in case the problem had artificially restricted the domain. But no, you have the natural domain (the largest domain that makes sense). So, based on that information, plus the corrected derivative, can you tell me what the domain of the derivative is?

#### purplec16

The overall limit is 0/h

But nope...i'm still confused with the domain

I have to go to class...i guess i'll ask my lecturer

#### Ackbeet

MHF Hall of Honor
The overall limit is 0/h
Incorrect. Once you've taken the limit, there should no longer be an h in it.

But nope...i'm still confused with the domain
Let's get the correct derivative before going on (see above).

#### Duke

The domain is obviously the set of real numbers. As for the limit,
$$\displaystyle \lim_{h \to \0}( 0/h)$$=$$\displaystyle \lim_{h \to \0}( 0/1)$$ by l'hopital's rule which can be used since 0/0 is an indeterminate form.

#### Ackbeet

MHF Hall of Honor
The domain is obviously the set of real numbers. As for the limit,
$$\displaystyle \lim_{h \to \0}( 0/h)$$=$$\displaystyle \lim_{h \to \0}( 0/1)$$ by l'hopital's rule which can be used since 0/0 is an indeterminate form.
Doubtful that l'Hopital's Rule is available at the moment, since it depends on derivatives, which is exactly what we're trying to compute.

#### Duke

$$\displaystyle \lim_{h \to \0}0/h$$ =$$\displaystyle \lim_{h \to \0}0$$

#### Ackbeet

MHF Hall of Honor
$$\displaystyle \lim_{h \to \0}0/h$$ =$$\displaystyle \lim_{h \to \0}0$$
I agree; this is what I was getting at in post # 4, and what I was trying to get the OP'er to figure out on her own.

"Excite and direct the self-activities of the pupil, and as a rule tell him nothing that he can learn himself." - Law # 6 from John Milton Gregory's The Seven Laws of Teaching.

• TheEmptySet and purplec16