Is $u(t)=u(2t)$ true?

Question

Since $u(t)=1$ for $t>0$ and $0$ otherwise, this means that $u(t)=u(2t)$ for all $t$. Also, they both have identical graphs.

But when obtaining the derivative of both of these functions they give different results. $$\frac{du(t)}{dt} = \delta(t)$$ $$\frac{du(2t)}{dt} = 2\delta(2t)$$ Where the second derivative is obtained using the chain rule.

So does this mean that $u(t)\neq u(2t)$? or the second derivative is wrong?

Answer 1

Hint
We want derivative in the sense of distributions. So to check equality, you need to check equality in the sense of distributions. What distribution is $2 \delta(2t)$? That is, if $\varphi$ is a test function, what is $\langle 2\delta(2t), \varphi\rangle$? Is that the same distribution as $\delta(t)$? Answer: Yes.

But: if you do not know how to compute in distributions, then this question is too advanced for you.