Thread: Chain Rule - Proof Using...

I have the question, and part of my work... I can't figure out where to go from here. Bear in mind, this is university level calculus, grade 12... We don't do this type of question too often, so I might need the extra step included in between your regular steps.

Givens:
y=f(x)
f(a)=a
f '(a)=b
g= f o f o f o f o f o f o f o f o f (if I'm using the wrong symbol here, I mean to say f at f at f at f at ....... and so on, 10 times)

Question: find g '(a)

Work/Answer:
g(x)=(f(x))^10
g(a)=(f(a))^10

Originally Posted by mike_302

I have the question, and part of my work... I can't figure out where to go from here. Bear in mind, this is university level calculus, grade 12... We don't do this type of question too often, so I might need the extra step included in between your regular steps.

Givens:
y=f(x)
f(a)=a
f '(a)=b
g= f o f o f o f o f o f o f o f o f (if I'm using the wrong symbol here, I mean to say f at f at f at f at ....... and so on, 10 times)

Question: find g '(a)

Work/Answer:
g(x)=(f(x))^10
g(a)=(f(a))^10

?? The problem asked you to find g'(a). Why did you give g(a)?

Try a simple case first. Suppose g(x)= f(f(x)). Then g'(x)= f'(f(x))f'(x). Then g'(a)= f'(f(a))f'(a)= f'(a)f'(a)= .

If g(x)= f(f(f(x))), then g'= f'(f(f(x))(f(f(x))' so .

Now, make a wild guess at what your answer should be!

Originally Posted by mike_302

I have the question, and part of my work... I can't figure out where to go from here. Bear in mind, this is university level calculus, grade 12... We don't do this type of question too often, so I might need the extra step included in between your regular steps.

Givens:
y=f(x)
f(a)=a
f '(a)=b
g= f o f o f o f o f o f o f o f o f (if I'm using the wrong symbol here, I mean to say f at f at f at f at ....... and so on, 10 times)

Question: find g '(a)

Work/Answer:
g(x)=(f(x))^10
g(a)=(f(a))^10

Define .

Observe a couple of facts:
1. for , and
2. for all .

So .

What is ?
Well it is . By the chain rule, this is .

We continue this process until the subscript is 1 in the rightmost term.

.

By observation 2 above, then:
.

Can you take it from there?

Ahhh! Clever. I did not think to start off with only the pair and go from there. I was trying to do it with all ten in there.

Okay, that's an interesting way of tackling it.

Appreciated!