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What is $T(p(a x^2+b))$ when $T(p(x))=x^2p'(x)$

Mathematics Asked by buttercrab on December 8, 2021

Let $T$ be linear transformation on $mathbb{P} rightarrow mathbb{P}$ that $T(p(x))=x^2p'(x)$.

1. Then, what is the value of $T(p(ax^2+b))$?

I think it could be $x^2p'(ax^2+b)$ or $2ax^3p'(ax^2+b)$ or even something else.

2. What would be the result when $T$ is on $mathbb{D} rightarrow mathbb{D}$? ($mathbb{D}$ is set of differentiable functions)

One Answer

$T(p(ax^{2}+b))=x^{2} frac d {dx} (p(ax^{2}+b))=x^{2}p'(ax^{2}+b) (2ax)$ by Chain Rule. The answer is same for $mathbb D$.

Answered by Kavi Rama Murthy on December 8, 2021

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