I'm trying to do some last-minute math revision, as I have a test on Monday, and I found a question I can't do. And the mark-scheme doesn't make sense.

The question:

f:x |-> 3^x

Find the solution of the equation f''(x) = 2

My first step was to find the inverse of f(x)... f'(x)

So I let k = 3^x
ln(k) = ln(3^x)
ln(k) = x(ln(3))
x = (ln(k))/(ln3)
(x = log(base 3)(k))

But then I tried finding the inverse of the inverse... f''(x)and got the original function.

The answer scheme, however, says that f'(x) is not what I got at all, but is actually (3^x)(ln(3)).

What am I doing wrong?

Well I'm not actually sure what you mean by inverse, what are trying to do with it, or why it would help you get f'(x). You are however correct that x = log(base 3)(k), but I would say that should be obvious from the initial problem, as it is basically the definition of a logarithm.

Here is how I would figure out f'(x) and f''(x):
f(x) = 3^x
f(x) = e^(x * ln(3))
f'(x) = d(e^(x * ln(3)))/dx
f'(x) = e^(x * ln(3)) * ln(3)
f'(x) = 3^x * ln(3)
f''(x) = d(3^x * ln(3))/dx
f''(x) = d(e^(x * ln(3)) * ln(3))/dx
f''(x) = ln(3) * d(e^(x * ln(3)))/dx
f''(x) = ln(3) * ln(3) * 3^x

[ 06-04-2005, 01:58 AM: Message edited by: Seraph ]

Dammit! That's where the problem lies! I knew I was forgetting something... f'(x) doesn't mean the inverse of x, it means the derivative of it!

D'oh!

Thanks! :D

[ 06-04-2005, 02:04 AM: Message edited by: Callum ]

and f''(x) is the second derivative

f-1(x) is the inverse i think.

Yeah... bodes well for monday's exam no? :D

the inverse is 1/f(x) and onviously, doing 2 inverses would be 1/1/f(x) which =f(x), hence it got Callum nowhere [img]smile.gif[/img]

[ 06-04-2005, 08:21 AM: Message edited by: Aragorn1 ]

Please tell me this is not Grade 10 math!! :'(

my math exam has been very easy so far, 25 minutes for 15 multiple choice questions [img]smile.gif[/img]

I'm in the 'year 13' although we wouldn't call it that. I'm at college, so i'm a second year. I think it used to be called 'upper-sixth,' but only the oublic schools call it that now i think.

Maths multiple choice!?

*blinks*

Wow.. Sounds... difficult.

And I'm in the year below Aragorn1... though we do call it year 12/13 here.

Does your school go from yr 7-13?