Simple Laplace transform question

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The discussion centers on understanding the Laplace transform of the first derivative of a function. The user is confused about how the term "sU" is derived in the context of the transform, as it is not found in their reference table. They specify that the function y(t) has an initial condition of y(0) = 0. The formula for the Laplace transform of the first derivative is clarified as L{y'(t)} = sY(s) - y(0), which simplifies to L{y'(t)} = sY(s) when y(0) = 0. This highlights the importance of initial conditions in deriving Laplace transforms.
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in an example in my text i don't see how they got the "sU" for the transform. actually, i don't even see it in my table of transforms.
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Say we have ##y(t)## with ##y(0)= 0##. What is the formula for the transform of the first derivative?
 

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