Trig derivatives

[losfomot]

Can someone help me on these two questions... you must find the first two derivatives of f(x)...

 

first:

 

f(x)= -cot(x)

 

I think I've got the first one...

 

f'(x)= csc^2(x)

 

but I'm lost on the second derivative.

 

Next Question:

 

f(x)= sec(x) - csc(x)

 

again, I think I've got the first one...

 

f'(x)= sec(x)tan(x) + csc(x)cot(x)

 

but f''(x)= ???

 

Thanks.

[Tom Mattson]

Can someone help me on these two questions... you must find the first two derivatives of f(x)...

 

first:

 

f(x)= -cot(x)

 

I think I've got the first one...

 

f'(x)= csc^2(x)

 

Yep.

 

but I'm lost on the second derivative.

 

Didn't Aretha Franklin have a song about this? Oh yeah' date='

 

[i']"Chain chain chain, the chain ruuuuule."[/i]

 

Ba-dum-bump.

 

Seriously, you can think of csc²(x) as [csc(x)]², and then use the chain rule.

 

Next Question:

 

f(x)= sec(x) - csc(x)

 

again, I think I've got the first one...

 

f'(x)= sec(x)tan(x) + csc(x)cot(x)

 

Yep.

 

but f''(x)= ???

 

You need the product rule this time, and you'll need to apply it to each term in f'(x).

[losfomot]

Didn't Aretha Franklin have a song about this? Oh yeah' date='

 

[i']"Chain chain chain, the chain ruuuuule."[/i]

 

Ba-dum-bump.

 

Seriously, you can think of csc²(x) as [csc(x)]², and then use the chain rule.

 

Thanks for the help Tom... I got both questions correct now, but I didn't use the chain rule, I used the product rule for both of them. Exactly how would you use the chain rule for the first question?

[jordan]

The product rule would work fine for [csc(x)]² but the chain rule would be f'(g(x))*g'(x). Perhaps you haven't reached this part yet, it usually comes shortly after the product/quotient rules.

 

Anyway, let csc(x)=g(x)

 

and f(x) would be g(x)²

 

If you follow the formula above this will also give the right answer, and often in less work than the product rule would for the same thing.

 

By the way, one thing that screwed me up about this was the f(x)=g(x)² part. Don't fill in for g(x) right away. Take the derivitive first and then fill in again for g(x).