Derivative of ln(cos5(3x4)): Using Chain Rule to Find Solution

[prime-factor]

Homework Statement

ln(cos⁵(3x⁴))

Homework Equations

chain rule:

y = f(g(h)

y' = f'(g(h) x g'(h) x h'

The Attempt at a Solution

y' = 1 / ((cos⁵(3x⁴)) x 5((cos⁴(3x⁴)) x -sin(3x⁴) x 12x³

=> -5sin(3x⁴)(12x³) / cos(3x⁴)

=> -60x³sin(3x⁴) / cos(3x⁴)

=> -60x³tan(3x⁴)

=>No further simplification that I can see

 

[NoMoreExams]

Why would you use x as multiplication when it's your variable...

 

[prime-factor]

Sorry
I'll know not to next time.
Thankyou

 

[NoMoreExams]

Maple confirms your answer :)

 

[prime-factor]

Thanks.

 

[NoMoreExams]

If you studied integrals you can always just integrate your answer and see if it's what you started with give or take a constant.

 

[prime-factor]

I'm just starting to learn very basic integration before I start the school year. I would not know where to start integrating that solution to check at this stage . Would you be able to explain a good approach to integrating it? Or should I post it as a problem after I've attempted it?

 

[NoMoreExams]

You should probably attempt it, there's a pretty basic u-subst. for this one

 

[Dick]

I have NEVER heard anyone suggesting that you use integration to check a differentiation. The reverse, yes. Don't do it. Differentiation is easy. Integration is hard. Sometimes even when you know the answer. If you want integration practice go ahead, but it's not a practical check. You are much more likely to make a mistake in the integration than the differentiation.

 

[NoMoreExams]

That may be true but this is a pretty simple integral :)

 

[prime-factor]

Regardless. I am going to practice more integration and eventually (hopefully) master it successfully. It is already proving to need more problem-solving skills than derivatives!

 

[Dick]

That may be true but this is a pretty simple integral :)

True, but then the differentiation isn't that hard either. I'm saying I don't recommend it as a general strategy. And I wouldn't worry the OP that being able to integrate is all that helpful in being able differentiate accurately.

 

[Dick]

Regardless. I am going to practice more integration and eventually (hopefully) master it successfully. It is already proving to need more problem-solving skills than derivatives!

Right you are.

 

[NoMoreExams]

I can't speak to that, I've always done it both ways (then again I always had Maple to check my answers also).

 

[Dick]

I can't speak to that, I've always done it both ways (then again I always had Maple to check my answers also).

That is a factor. But why not just use Maple to differentiate it then?

 

[prime-factor]

You two seem two know what you're talking about so I'd like to ask this:

I have a book by Kaplan and Lewis called: "Calculus and Linear Algebra [combined edition]. It is quite old (1970).

Is that a good book in your opinion (assuming you know it or the authors works?

 

[NoMoreExams]

Dick: because in my career sometimes I don't have the luxury of having Maple

Prime-factor: I've never heard of it, I used Stewart to learn Calc. 1-3

 

[Dick]

I don't know the book, either. Old ones aren't necessarily bad. Maple may be a luxury, but you don't need it. I went through my grad career using Mathematica. I was really quite hooked. Since I don't have people buying me CAS software anymore, I switched to Maxima. It's really pretty good. I can live with it. And it's FREE. But Prime-factor, if you do start using something like that, always make sure you know how to get the answer without the software, ok?

 

[prime-factor]

No problem there. I'm a firm believer in practicing it 100 times if I don't get the answer before using a calculator. I've fallen into that trap once before and never again!. That's why I have plenty of Sharpies and A4 paper at my desk at all times, I do practice everything I learn, plus I'm starting year 12 this year with 2 Maths Subjects, so I've basically learned the stuff I'll need for next year (except integration) in a couple of months from watching online tutorials and reading and hundreds of A4 pages of practice:P.

 

[Dick]

You will do so fine.