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Impulse response of a causal LTI system without using Laplace transform

The problem with your first approach is that you assume $$f(t)\delta'(t)\stackrel{?}{=}f(0)\delta'(t)\tag{1}$$ which is wrong. The correct equation is $$f(t)\delta'(t)=f(0)\delta'(t)-f'(0)\delta(t)\...
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