Derivatives Exercises

Answer

We have a = 0 and f(x) = x², so we can get right to the limit:

Looks like f ' (0) = 0.

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.

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.

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.

Answer

With a = 1, we have

After all that, we can say  

Answer

We start with a = 0, then

And there's the derivative in all it's glory.

Answer

Beginning with a = 0, we can move onto the limit.

Uh-oh! As h approaches 0, so does h^2/3. This means  approaches ∞, so