Domain of a Derivative?

Question

How do you find the domain of a derivative. I did a problem where i found the derivative of f(x)=.5x-1/3 and i got 1/2. Now how do i find the domain of "1/2".

Do I find the domain of the tangent line which has a slope of 1/2?

Thanks!

Thanks but that doesn't help me. Show me how to catch the fish instead of giving me the fish.

Answer 1

The derivative of a function is itself a function. As you stated, the derivative of f(x)=(1/2)x+1/3 is f'(x)=1/2. The domain of this function is all real numbers.


Examples:
If the function is f(x)=1/x. Then f'(x)=-1/(x^2). The original function has domain all reals except x=0 (or (-infinity,0) U (0,infinity))
The derivative also has this domain.

Very often the function and its derivative will have the same domain, but sometimes not:

If f(x)=x^(1/3) (cubed root), then f(x)=(1/3)x^(-2/3).
f has domain all reals, but f' is not defined at zero.


Note that the derivative of f(x) at x=c is given (if it exists) by the limit

lim (x-->c) (f(x)-f(c))/(x-c).

So we use f(c) in the definition. Therefore for the derivative to be defined at a point, the function must be defined there also. We say this as "the domain of the derivative is a subset of the domain of the function."


Hopefully this helps you wield that rod (fishing that is).

Answer 2

The domain of the (first) derivative of a function is the union of the domains of function and the domain of the derivative function. In this case, both the function (f(x)=x/2-1/3) and derivative function (f'(x)=1/2) have the entire set of real numbers as their domains, so that is the domain of the (first)derivative of f(x).

However, it is not always the case that the derivative function and the function have the same domain- look at f(x)=ln(x). In this instance, f'(x)=1/x, which has domain R-0 (the reals without 0), wheres ln(x) has domain R+ (positive real numbers). In this case the domain of the derivative function is R+.

Answer 3

This is a linear function and the derivative is a constant
(it has the the same value for each x real)

Answer 4

by-product: Step a million: x^a million (there is an invisible exponent of a million for x) x^a million -> placed it because of the fact the backside 1x -> the x cancels out because of the fact you subtract a million from the exponent (and all others to boot once you're searching for by-product) -> 1x^a million-a million -> a million =a million Step 2: x^a million/2 a million/2x^-a million/2 x^-a million/2 ability sq. root because of the fact the invisible a million in front of the x can no longer cancel something out --> a million/2?x a million+(a million/2?x)