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anonymous
 one year ago
How do I find the derivative of...
anonymous
 one year ago
How do I find the derivative of...

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anonymous
 one year ago
Best ResponseYou've already chosen the best response.0\[f(x)=\frac{ 5 }{ (2x)^3 }+2cosx\]

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1A rule of exponents let's us write it like this: \(\large\rm f(x)=5(2x)^{3}+2\cos x\)

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1Differentiating the first term shouldn't be too bad, power rule into chain rule, ya?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I am horrible with trigonometric identities though, so I'm not sure what to do with the cos part.

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1Woops:\[\large\rm \color{royalblue}{\left[5(2x)^{3}\right]'}=3\cdot5(2x)^{4}\cdot\color{royalblue}{(2x)'}\]Forgot your chain rule I think.

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1\[\large\rm =3\cdot5(2x)^{4}\cdot(2)\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I'm a bit unfamiliar with the rule... So you would take the 2 from (2x) and multiply it by the rest?

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1You learn these tricks early on: Power Rule Product Rule Quotient Rule Chain Rule Chain Rule is the most difficult of these to master, by far. You'll need to do a lot lot lot of practice problems to feel comfortable with it.

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1You're always multiplying by the derivative of the `inner function`. So yes, for this problem, the inner function is 2x. We have to multiply by the derivative of 2x on outside like that.

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1Another example: \(\large\rm 5(x^2+3)^5\) I apply the power rule to outermost function which is \(\large\rm 5(\qquad\quad)^5\) giving me \(\large\rm 5\cdot5(\qquad\quad)^4\) Chain rule tells me that I have to multiply by the derivative of the inner function.\[\large\rm 5\cdot5(\qquad\quad)^4\cdot\left(\qquad\quad\right)'\]Where ( ) is whatever was inside of the outerfunction that we had. The prime says to take a derivative. \[\large\rm 5\cdot5(\qquad\quad)^4\cdot (2x+0)\]So I multiplied by the derivative of x^2+3, the inner function. So the final result is:\(\large\rm 5\cdot5(x^2+3)^4(2x)\) Just a silly example :O I hope the empty brackets didn't make it MORE confusing lol

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1For the trig stuff.. honestly, just memorize them for now. Sine and cosine are very cyclical. When you differentiate sine a number of times, it follows this pattern: \(\large\rm \sin x\quad\to\quad \cos x\quad\to\quad \sin x\quad\to\quad\cos x\quad\to\quad\sin x\) So if you differentiate sin x, you get cos x. If you differentiate cos x, you get sin x. If you differentiate sin x `four times`, you get right back to sin x! :)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Hey I'm back. Sorry, my internet gave out for a bit. It tends to do that every now and then.

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1And keep in mind that constant coefficients have effect on the differentiation process. Just carry the 2 along for the ride.\[\large\rm 2\sin x\quad\to\quad 2\cos x\quad\to\quad 2\sin x\quad\to\quad2\cos x\quad\to\quad2\sin x\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I see how in your "Another example" you got the 2x from x^2, but why did the 3 turn into a zero?

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1Differentiation is all about measuring "change". Something that is `constant` by definition, does not change. So when I'm taking a derivative, this "dx" represents an immeasurably small change in x. So when d/dx looks at the 3, he's asking the question "when x changes a very small amount, how much does 3 change?" And the answer is 0 amount. It's rate of change is 0. If you prefer, you can think of it in terms of power rule:\[\large\rm \frac{d}{dx}3\cdot\color{orangered}{1}=\frac{d}{dx}3\cdot\color{orangered}{x^0}=3\cdot0x^{1}=0\]

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1Recall that anything to the zero power is just 1. \(\large\rm a^0=1,\qquad \forall a\ne0\). So I used this clever idea to rewrite 1 as x to the 0.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0So any number besides 3 would also turn into zero as well?

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1Yes. And later on, they might try to trick you and give you "fancy numbers" as I like to call them.

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1\[\large\rm \left(e^{4\pi}\right)'=0\]There is no variable! The whole thing is just a constant!

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Ah, okay. That seems simple enough.

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1Polynomials will follow this progression as you differentiate them:\[\large\rm cubic\quad\to\quad square\quad\to\quad linear \quad\to\quad constant\quad\to\quad 0\]

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1That's just an illustration of the power rule ^ Obviously higher power follow it as well :)

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1Bahh I gotta run to the grocery store before it closes :O You got this figured out? XD

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I believe so. Oh quick question. I just convert cosine to sine right?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0If that will take too much time to explain, then don't worry about it. I don't want to be the cause of you not having food. :)

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1Good! :D And the 2 comes along for the ride of course.\[\large\rm f(x)=5(2x)^{3}+2\cos x\]\[\large\rm f'(x)=3\cdot5(2x)^{4}(2)2\sin x\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0You're better at explaining this than my textbook or teacher.

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1Ehh it's easy to say that when you're getting 1 on 1 attention XD lol Gotta give teacher just a little bit of slack haha

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Well, I have in the past but eh. I guess I just have a different learning style. Good night! :)
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