Question

Conjecture Let \(f\) be a differentiable function of period \(p .\) (a) Is the function \(f^{\prime}\) periodic? Verify your answer. (b) Consider the function \(g(x)=f(2 x) .\) Is the function \(g^{\prime}(x)\) periodic? Verify your answer.

Short answer

Yes, the derivative of a periodic function is also periodic. The derivative of a function \(g(x) = f(2x)\) is only periodic if the period of the derivative of \(f\) is twice the period of \(g\).

Step-by-step solution

Determining if the derivative of a periodic function is periodic

A function \(f(x)\) is said to be period if there exists a positive real number \(p\) such that for all \(x\) and all integers \(n\), \(f(x+n p)=f(x)\). If \(f\) is differentiable, then the derivative \(f'(x)\) would have the property \(f'(x+np)=f'(x)\), for any integer \(n\). Thus the derivative of a periodic function is also periodic.

Setting up the function \(g(x)=f(2x)\) and its derivative

The function \(g(x)\) is a transformation of \(f(x)\), where we replace \(x\) with \(2x\). We can differentiate \(g\) with respect to \(x\) to get: \[g'(x) = (f(2x))' = 2f'(2x) \] because of the chain rule of differentiation.

Determining if \(g'(x)\) is periodic

The derivative function \(g'(x)\) is periodic if there exists a \(p > 0\) such that \(g'(x + p) = g'(x)\), for all \(x\) and all integers \(n\). Given our definition of \(g'(x)\), we find \(g'(x + p) = 2f'(2x + 2p)\), which by the periodicity of \(f'(x)\), is equal to \(2f'(2x) = g'(x)\) if \(2p\) is a period of \(f'(x)\). Thus \(g'(x)\), in this case, would only be periodic if \(p\) is equal to half the period of \(f'\).

Conclusion

The derivative of a periodic function \(f'(x)\) is also periodic with the same period as \(f\). However, for a function \(g(x)\) defined as a transformation of \(f(x)\), its derivative \(g'(x)\) is periodic only if the period of \(f'(x)\) fits the transformation rule.