Proving one Differentiation results to another

1. Oct 27, 2008

Khayyam89

1. The problem statement, all variables and given/known data
I need in proving that the derivative (d$$^{n}$$/dx$$^{n}$$)(sin4x + cos4x) = 4n-1 cos(4x + n$$\pi$$/2)

3. The attempt at a solution
I understand implicit differentiation in basic problems but I get stump with the n exponent in the differentiation symbol; am I suppose to treat it as a 2nd, 3rd, 4th ... etc derivative?
If thats so, how should I prove that the left equation equals the right one.

So far I got to: 4(cos3x - sin3x)

2. Oct 27, 2008

rock.freak667

If you want you could try proving it by mathematical induction.

dn/dxn means the nth derivative

3. Oct 28, 2008

HallsofIvy

Staff Emeritus
For the "induction step" you need to prove that IF
$$d^n 4(sin^4 x+ cos^4 x)= 4^{n-1} cos(4x+ n\pi/2)$$
then
$$d^}{n+1} 4(sin^4 x+ cos^4 x)= 4^{n} cos(4x+ (n+1)\pi/2)$$
You should be able to do that just by differentiating the right hand side of the first equationl.

4. Oct 28, 2008

Khayyam89

Thank you, I understand now.