The first thing you need to recognize is that the first part of the equation is multiplying a variable times a variable. You need to use the product rule.

Here is the rule:

u'v + uv' What

Comes

Next? Time 4 "Log"

Rolling! The Perfect Answer! You are taking a Calculus

test. All of a sudden you stop. Fear grips your soul. Oh No! A complicated Derivative! What do you do? Wipe up your tears. Here's your answer! If you got that, then you got the hardest part. Tears dry now? Good. Let's see what you got. Here's my part one.

6(sec4x)^2 4(csc4x)^2 + 6tan4x 2csc4xcsc4xcot4x

4x

Now, that seems just a bit long. Why don't we simplify that? After all, that IS a lot of numbers. Here's what I got.

24(sec4x)^2 (csc4x)^2 + 48tan4x(csc4x)^2 cot4x

Be Clever! Do you see the simplification I left out? Hopefully, you did. See that cotangent? He's your friend.

24(sec4x)^2 (csc4x)^2 + 48(csc4x)^2

POOF! He's Gone! What about the other part? By now we have done the entire problem. We just have them in 2 pieces. Now what to do? Smash them together. Your answer should look something like this...

24(sec4x)^2 (csc4x)^2 + 48(csc4x)^2 - 2 / x

Look! Your treasure

awaits! ----------->

See? Annoying,

but not so scary! Derivatives! A How-To on how to solve

those hairy derivatives

that make you cringe! Through your blurry eyes, you see the problem.

6tan4x(csc4x)^2 - lnx^2

Don't panic! It really isn't that hard once you look

at it. We just need to break it up.

(I'd write the problem down!) Part One: The Product Rule What Now? So, we've solved part one. The scary part is done. Compose yourself and we can continue. Ready? Okay. If that last part was like trying to wash a rabid, angry feline (with claws I might add), this part is like trying to wash a big, happy Labrador who loves the water. WAYYYYYY easier. Now we need to do the remaining part of the equation. What was it again? Oh yeah. Something like this:

-(subtraction) ln2x^2

Can you do this? Come on it's easy. Don't forget your chain rule. Give it a try while I just stand here and stare at you while I wait. It's okay, just take your time.... All the time you need... Problem "u" represents the first function, in this case 6tan4x. "v" represents the second function, in this case (csc4x)^2. Try using the product rule, along with the chain rule, to complete the first section of this function. You'll see my work in just a moment. (Don't be scared! Functions don't bite!) What do you get? Finally! Wasn't that hard? I really hope you can hear the sarcasm in my voice. Okay. Now I can show you what I got. Here it is.

- 1 / 2x^2 4x

Hopefully you remembered the chain rule? I hope so! However, this simplifies. Simplify it and you get - 4x / 2x^2. Can you simplify it further? Please say yes. Because if not, then I'm the one who looks stupid saying that it simplifies to - 2 / x. But I'm pretty sure I'm good.

Ready to go? Let us proceed. "Knowledge is our greatest treasure"

What a rip-off! Thanks for putting up with Me! Hopefully you won't have to do derivatives this bad ever again, but if you do I'm sorry. But think of it this way. If you can manage to pull this off, you should have absolutely NO problem with the wimpy little derivatives. Try not to cry too hard if you see a problem this big on a test. I probably will, but hopefully you won't! Happy Differentiating!

(Is that even possible???)

(And if you do them then..........)

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# AP Calculus Derivative Project

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