Using the limit definition, what is f ' (0) if f(x) = x2?
Answer
We have a = 0 and f(x) = x2, so we can get right to the limit:
Looks like f ' (0) = 0.
Example 2
Using the limit definition, what is f ' (-1) if f(x) = 1 – x2.
Answer
We have a = -1 and f(x) = 1 – x2, so going to the limit, we find
This next step is a dangerous one. When evaluating f(-1 + h), make sure to plug in all of (-1 + h) for x, and don't forget the rest of the function f.
There it is: f ' (-1) = 2.
Example 3
Using the limit definition of the derivative, what is f ' (1) if f(x) = x3?
Hint
Yes, we do need to cube (1 + h).
Answer
We have a = 1 and f(x) = x3. Now plug this into the limit definition of the derivative:
Since (1 + h)3 is not quite a barrel of monkeys, we need to work it out separately here and then put it back in the formula. This requires the Distributive Law and combining like terms.
Now back we go:
Therefore f ' (1) = 3.
Example 4
If it exists, use the limit definition to find the derivative of f ' (-1) if f(x) = x2 – x.
Answer
Our value for a is -1. With that out of the way, we can get to it.
Since this one's a bit tricky, we'll find f(h - 1) and f(-1) separately.
And now put them back in:
Therefore f ' (-1) = 3.
Example 5
Using the limit definition to find the derivative, if it exists, what is f ' (1) if ?
Answer
With a = 1, we have
After all that, we can say
Example 6
Using the limit definition to find the derivative, what is f ' (0) when ?
Answer
We start with a = 0, then
And there's the derivative in all it's glory.
Example 7
Using the limit definition to find the derivative, what is f ' (0) when f(x) = x1/3?
Answer
Beginning with a = 0, we can move onto the limit.
Uh-oh! As h approaches 0, so does h2/3. This means approaches ∞, so
?
?
approaches ∞, so 