Derivative Mnemonics

[magicdanw]

Hi. I've got a math exam tomorrow, without calculators, and I'm having trouble memorizing the most recent derivative rules we've learned. These are the rules for inverse trig functions, LOGaU, and a^u. Does anyone have any useful mnemonics for remembering these, or any methods of deriving them from the other derivative rules? Thanks!

[DarkerLine]

You can derive the inverse trig derivatives using implicit differentiation (which you may not have learned yet, but it's quite simple as you'll see). For example, here is how you derive the derivative of arctan:

Let y = arctan x. We want to know dy/dx.

tan y = x

d/dx (tan y) = 1

1/(cos² y) dy/dx = 1

dy/dx = cos²y = cos²(arctan x)

This is the correct derivative, but it can be simplified (although it's tricky). We need to find the cosine of the arctangent of x. Draw a right triangle with legs 1 and x. Then the tangent of the angle opposite x is x/1 or x, so that angle is the arctangent of x. Now find the cosine of this angle: this is the adjacent leg over the hypotenuse, which by the Pythagorean Theorem is √(x²+1), so the cosine is 1/√(x²+1). The cosine is squared in the derivative we found, so we square this quantity for our final answer: dy/dx = 1/(x²+1)

As for the other two types of functions you mentioned, all you need to know is that the derivative of ln x is 1/x, and the derivative of e^x is e^x. Then other bases will follow easily if you write log_ax as (ln x)/(ln a), and a^x as e^{x ln a}.

[simplethinker]

I'm drawing up some (crappy) pictures, so just wait a couple min :biggrin:

[edit]
does the bottom side look familiar?

[edit 2]
square the hypotenuse, does it look familiar?

@DarkerLine: try figuring out the derivatives of inverse trig funtions using the euler identities, so sin(x)=(e^ix-e^-ix)/(2i) and cos(x)=(e^ix+e^-ix)/2
(so you have to define the inverse trig functions in terms of the natural log, just like the inverse hyperbolic functions) it's fun :biggrin:

[magicdanw]

Thanks for the explanations and the not-so-crappy pictures, DarkerLine and simplethinker! I'll practice writing them out to make sure I can remember the derivations. I've always been better at understanding concepts than memorizing things by rote, so this will help a lot! :biggrin:

Update: Yup, I just took out a blank sheet of paper and correctly derived arcsin. I really wish my teacher taught the class how to derive these instead of telling us to just memorize them.

[simplethinker]

Nah! figuring out how to derive them yourself is 1) more fun, and 2) better for long-term retention in your memory!! (or maybe that's just me, since I guess you could call me a nerd)