In the previous lesson, we covered what a derivative is, and how to find it.
We learned that the standard formula to find the derivative of a function is
If you're unsure on how to type something here, check out some of the great tutorials on the internet! (Here's a good reference). If you want to see how I typed something, you can right-click on any of the math, and check out the TeX commands. Otherwise, Google is always your friend.
Remember your formula?
Try to use some of the identities we learned last chapter.
Distribute, and split the limit:
Look! There are our old friends,
LI4, hiding out in our practice problems! We can simply substitute, and move on:
More neatly written,
This question is solved almost the exact same way as the previous.
Remember that the limit operation is distributive across multiplication, division, addition, and subtraction. Below, we distribute amongst subtraction.
The value of is independent of , the approaching variable, so we can remove values that are functions of to be a coefficient of the limit.
Aha! There's our
Try to use
Set up the limit:
Now, we can factor out of the limit:
LI2! We can substitute
This is one of the most important differentiations, next to and . This will be particularly useful in the coming chapters.
What was the formula for ?
Apply the difference of logarithms formula:
Now we apply the logarithm coefficient formula:
Wow! Doesn't it resemble
However, the actual approaching variable is , not . So, we can't immediately substitute for our identity.
So, I guess we're stuck now- there's nothing we can do. Or is there?
Let's step away from the problem and think about it for a moment. As approaches 0, what does approach?
We can just plug in 0 to the numerator, and we get 0! So now, we can say that
(Make sure you see why that step was necessary.)
Now, we can just substitute