Solving for Higher Order Derivatives: Explaining f(x)=cos2x and f^50 = f^2

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SUMMARY

The discussion focuses on the higher order derivatives of the function f(x) = cos(2x) and the assertion that f^50 = f^2. The reasoning provided is based on the periodicity of the cosine function, specifically that the derivatives of cos(2x) repeat every four derivatives. The equation 50 = (12)(4) + 2 is used to demonstrate that the 50th derivative corresponds to the 2nd derivative, confirming that f^50 = f^2 due to this cyclical nature.

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Homework Statement



if f(x)=cos2x, how do we know that f^50 = f^2 ??

In the solution it states that because 50 = (12)(4) + 2, then f^50 = f^2.

What the??
 
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We don't know that f(50) = f(2) for the reason that it isn't true. If the function were f(x) = cos(x), it would be true, though, and this is easy enough to verify by taking a few derivatives and noticing the cyclical nature of these derivatives.
 
Okay, well how do they know that f^2 = f^50 for cosx simply by going 50 = (4)(12) + 2
 
Because it repeats every four times
cosx
-sinx
-cosx
sinx
cosx
 

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