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On functions f,g satisfying (fg)'=f'g'

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July 7, 2020
functionreal analysisdifferentiability

Problem Statement

For two real intervals I,J, I,J, we say that two functions f,g:I⟶J f,g:I\longrightarrow J have property P \mathcal{P} if they are differentiable and (fg)′=f′g′. (fg)'=f'g'.
a) Provide example of two nonconstant functions a,b:R⟶R a,b:\mathbb{R}\longrightarrow\mathbb{R} that have property P. \mathcal{P} .
b) Find the functions λ:(2019,∞)⟶(0,∞) \lambda :(2019,\infty )\longrightarrow (0,\infty ) having the property that λ \lambda along with θ:(2019,∞)⟶(0,∞),θ(x)=x2019 \theta :(2019,\infty )\longrightarrow (0,\infty ), \theta (x)=x^{2019} have property P. \mathcal{P} .
Dan Nedeianu
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