Two young mathematicians think about “short cuts” for differentiation.

Check out this dialogue between two calculus students (based on a true story):
Devyn
I hate the limit definition of derivative. I wish there were a shorter way.
Riley
I think I might have found a pattern for taking derivatives.
Devyn
Really? I love patterns!
Riley
I know! Check this out, I’ve made a chart So maybe if we have a function
Devyn
Hmmm does it work with square roots?
Riley
Oh that’s right, a square root is a power, just write So a square root is of the form .
Devyn
Let’s check it. If ,
Riley
Holy Cat Fur! It works! In this case .
Devyn
I wonder if it always works? If so I want to know why it works! I wonder what other patterns we can find?

The pattern holds whenever is a constant. Explaining why it works in generality will take some time. For now, let’s see if we can use the problem to squash some derivatives with ease.

Using the pattern found above, compute:
Using the pattern found above, compute:
Using the pattern found above, compute:
Using the pattern found above, compute: