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Find derivative of Just tell me 1-2 steps I will clip to them afterwards

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product rule within chain rule.
\[\large y=x^3 \sin^2(5x)\]We start by setting up the product rule.\[\large y'=\color{royalblue}{(x^3)'}\sin^2(5x)+x^3\color{royalblue}{(\sin^2(5x))'}\]
I already got to that point @zepdrix that's where I am stuck :D 2x^2sin(5x)+2x^3sin(5x)

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I forgot a power of 2 on the first sin *
\[\large y'=\color{orangered}{3x^2}\sin^2(5x)+x^3\color{royalblue}{(\sin^2(5x))'}\]I've taken the derivative of the first one. I think your coefficient is a lil mixed up. You brought the wrong exponent down :O
Bro which one are you referring to
The first blue term in the original product rule that I wrote out.
That? Am I not supposed to take down the power and multiply it with whatever it is next to it and and substract the power by 1?
x^3. Bringing the power down and subtracting by one gives us, 3x^2 right? :o
I am talking about the power of sin not x
Remember now we differantiating the second part of the expression or whatever
I wasn't talking about that one. I was doing the other term.
The second one will be a lil more complicated, I wanted to make sure you got the first part.
ah yea you got a point :D That one I didnt notice :D ok
So for the second one, again we'll apply the power rule to the sine function, then we'll apply the chain rule, multiplying by the derivative of the inside. \[\large y'=\color{orangered}{3x^2}\sin^2(5x)+x^3\color{orangered}{(2\sin(5x))}\color{royalblue}{(\sin(5x))'}\]The new blue term that showed up is the one we need to take a derivative of. If that is confusing, there is another way we can write it.
What we do this only for the second part if I may ask
Just need to understand whats the difference that forces us to do so
So we start with the product of two things involving x. \[\large y=(x^3)(\sin^2(5x))\] Without taking any derivatives yet, we can setup the product rule. \[\large y'=\color{royalblue}{(x^3)'}\sin^2(5x)+x^3\color{royalblue}{(\sin^2(5x))'}\] The blue terms are the ones we have to differentiate (take derivatives of). So you can see in our setup, we only differentiate the x^3 in the first term, and the sine in the second term. Was that the question? :o It's because of the product rule.
No I mean after that we do we take chain rule for Only the second term? Chain rule appears to be a lil messed up in my mind
Yah the second blue term has an inner function. Umm I'm trying to think of a good way to explain it...
if I have this (y5x)^2 Will I take chain rule for derivative?
(5x)^2 *** ignore the fact that I can just do it 25x^2 lets say I COULDNT. Will I use chain rule?
Yes that's a very good example :) Here's how we would work it out.\[\large \left[(5x)^2\right]' \qquad = \qquad 2(5x)\]That's the derivative of the OUTER FUNCTION. (Something squared). Now we have to multiply by the derivative of the inside.\[\large \left[(5x)^2\right]' \qquad = \qquad 2(5x)(5x)' \qquad = \qquad 2(5x)(5)\]
The chain rule can be really tricky to get a handle on. Lemme know if you need another example.
I see :) Alright I got it. Just something last because according to what I learned now I still cant make the solution like that in the solution manual (unofficial manual) maybe the above link is wrong?
No those steps are correct but they made a substitution to try and get the point across. I find that more confusing personally.. hmm
My own solution is 3x^2sin^2(5x)+2x^3sin(5x)cos(5x)
oh sh*t there is a 5 too hold on
Yes very good :) So you applied the chain rule TWICE, yes? After you got a cosine, you took the derivative of the (5x)
but still the manual doesn't end up with any cos
What they did in the solutions manual is, they applied the `Double Angle Formula for Sine`. That's why it looks a bit different.\[\large 2\sin x \cos x = \sin 2x\]So if you look at your second term, ignoring the x^2 in front, we have something like this,\[\large 5\cdot2 \sin(5x)\cos(5x)\]Applying the double angle formula gives us,\[\large 5 \cdot \sin(10x)\]
They did have a cosine if you look at the middle steps. They applied an annoying trig rule to get rid of it though :)
ooh so thats why
Not important for my course, im glad my way is correct too, thanks again
np c:

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