Proving one Differentiation results to another

[Khayyam89]

Homework Statement

I need in proving that the derivative (d$$^{n}$$/dx$$^{n}$$)(sin⁴x + cos⁴x) = 4^n-1 cos(4x + n$$\pi$$/2)

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 that's so, how should I prove that the left equation equals the right one.

So far I got to: 4(cos³x - sin³x)

 

[rock.freak667]

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

dⁿ/dxⁿ means the nth derivative

 

[HallsofIvy]

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.

 

[Khayyam89]

Thank you, I understand now.