I reached the chapter on Natural Logarithms in a calculus book I got from the library and I just don't understand how to differentiate ln. I try to solve them but the book gives a different answer. If anyone could explain how to differentiate the following equations, I'd appreciate it.

1) y = b(e^ax-e^-ax)
2) y = (3x² + 1) e^-5x
3) y = (3x² -1)(sq(x) + 1)
4) y = ( ln (x+3) ) / (x+3)
5) y = (a^x)(x^a)

I googled it but it didn't help, and the calc teacher was too busy. I figured someone here could help since almost everyone is a math wiz.

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Are you remembering your chain rules and multiplication rules?

Yes I did try those rules but I can't get them to work. There are a few other problems that I did solve correctly but those are the five that I've been trying to solve with no success.

Note to all: use the [sub] and [sup] tags!

You do realize that only one of those deals with natural logarithms? You haven't been thinking that e^whatever means that ln is involved somehow, have you?

They are solved using natural logarithms. You can't solve them using the normal rules.

1) y = b(e^ax-e^-ax)
// Just use the chain rule on each part in the ()
// You get a*e^x and -a e^-a*x
// Factor out the a and viola
y'=ab(e^ax+e^-ax)
2) y = (3x² + 1) e^-5x
// Use your multiplication rule d/dx(f*g)=f'*g+f*g'
// Simplify
y'=-15e^xx²-5e^x+6xe^-5x
3) y = (3x² -1)(sq(x) + 1)
// Another Multiplication rule
y'=15/2*x^3/2-x^-1/2/2+6x
4) y = ( ln (x+3) ) / (x+3)
// Division rule d/dx(f/g)=(f'*g-f*g')/g²
y'=(1-ln(x+3))/(x+3)²
5) y = (a^x)(x^a)
// Multiplication rule (see below for a^x)
y'=(a^x)(ax^a-1)+(ln(a)a^x)(x^a)

d/dx(a^x)=ln(a)a^x

That should be everything. Let us know if you need anything else explained.

Ah, I was shown to be wrong. Only 2 involve natural logs. I didn't know that d/dx(a^x) rule, though.

I didn't know that rule either. Also the answers you gave me for three and five disagree with the answers provided by the book. The book gives answers but not explanations. Here are the book answers for three and five:

3) (15x² + 12xsq(x) - 1)/(2sq(x))
5) a^x(ax^a-1 + x^a ln a)

The others were really helpful though. I was so frustrated with them I overlooked those rules. Thanks a lot.

In 3) they just factored out a x^-1/2/2 and in 5) they factored out an a^x. The answers are the same.

Ok cool. I have to get used to factoring stuff out... Thank you very much for the help. They were bothering me for the whole weekend.

No problem. I always hated the factoring out of things if it was nicer unfactorized (though they are nicer here).

I tend to factor stuff out as much as possible; the problem is I get a little carried away with it sometimes.