Nth Order Derivative of f(x): Sin^4(x)+Cos^4(x) & x^n/(1-x)

[shrody]

Homework Statement

The exercise goes "Determine d^n*f/dx^n for the 2 exercises:"
a) f(x)=sin^4(x) + cos^4(x)
b) f(x)= x^n/(1-x)

Homework Equations

The Attempt at a Solution

The only idea i had was for the second example, where i think its right to use a rule from Leibniz but I'm not sure...

 

[Mentallic]

I've never done any of these before, so please forgive me if I'm way off and not being helpful at all.


a) couldn't you express sin^4x and cos^4x in terms of multiple angles without powers, and then take the derivatives of those?


b) you can make the fraction like this: \frac{-(1-x^n)+1}{1-x}

and then they split up as so: \frac{-(1-x^n)}{1-x}+(1-x)^{-1}

The first fraction can be... expanded (for lack of a better word) into -(1+x+x^2+...+x^{n-1}) and I'm sure it's clear what to do with the second fraction.

 

[shrody]

i don't know about the first one with the angles, i think the whole point of the exercise is to keep the sin and cos and the second one kinda confused me...

 

[Mentallic]

Well you would still keep the sin and cos, just that theyre expressed as

sin^nx=Asin(nx)+Bsin((n-1)x)...

I know you could find the relationship easily for the n-th derivative from that, but if that isn't allowed, then maybe this will help?

y=sin^4x

\frac{dy}{dx}=4sin^3xcosx

\frac{d^2y}{dx^2}=-16sin^4x+12sin^2x

Notice how the sin^4x appears again. Maybe it's something, probably it's not, but I'm just putting that out there in case it helps.


For the second, is that "confused me" or "still confuses me"?

(x^n-1)=(x-1)(x^{n-1}+x^{n-2}+...+x+1)

dividing both sides by x-1 will show you how to get the long expansion thing.

Was this the problem?